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Existence of positive solutions for a critical nonlinear Schrödinger equation with vanishing or coercive potentials

Shaowei Chen

Author Affiliations

School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, P.R. China

Boundary Value Problems 2013, 2013:201  doi:10.1186/1687-2770-2013-201


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/201


Received:2 May 2013
Accepted:22 August 2013
Published:8 September 2013

© 2013 Chen; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we investigate the existence of positive solutions for the following nonlinear Schrödinger equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M1">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M3">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M4">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M9">View MathML</a>.

MSC: 35J20, 35J60.

Keywords:
semilinear Schrödinger equation; vanishing or coercive potentials

1 Introduction and statement of results

In this paper, we consider the following semilinear elliptic equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M10">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M11">View MathML</a>. The exponent

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M12">View MathML</a>

(1.2)

with the real numbers b and s satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M13">View MathML</a>

(1.3)

By this definition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M14">View MathML</a>.

With respect to the functions V and K, we assume that

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M15">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M18">View MathML</a>.

(A2) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M20">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M21">View MathML</a>

(1.4)

A typical example for Eq. (1.1) with V and K satisfying (A1) and (A2) is the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M22">View MathML</a>

(1.5)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M23">View MathML</a>, the potentials are vanishing at infinity and when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M24">View MathML</a>, the potentials are coercive.

Equation (1.1) arises in various applications, such as chemotaxis, population genetics, chemical reactor theory and the study of standing wave solutions of certain nonlinear Schrödinger equations. Therefore, they have received growing attention in recent years (one can see, e.g., [1-6] and [7-10] for reference).

Under the above assumptions, Eq. (1.1) has a natural variational structure. For an open subset Ω in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M26">View MathML</a> be the collection of smooth functions with a compact support set in Ω. Let E be the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M27">View MathML</a> with respect to the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M28">View MathML</a>

From assumptions (A1) and (A2), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M29">View MathML</a>

are two equivalent norms in the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M30">View MathML</a>

Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M31">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M32">View MathML</a>

Moreover, assumptions (A1) and (A2) imply that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M33">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M34">View MathML</a>

Then, by the Hölder and Sobolev inequalities (see, e.g., [[11], Theorem 1.8]), we have, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M35">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M36">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37">View MathML</a> is a constant independent of u. It follows that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M38">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M39">View MathML</a>

This implies that E can be embedded continuously into the weighted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M40">View MathML</a>-space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M41">View MathML</a>

Then the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M42">View MathML</a>

is well defined in E. And it is easy to check that Φ is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M43">View MathML</a> functional and the critical points of Φ are solutions of (1.1) in E.

In a recent paper [12], Alves and Souto proved that the space E can be embedded compactly into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M44">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M46">View MathML</a> and Φ satisfies the Palais-Smale condition consequently. Then, by using the mountain pass theorem, they obtained a nontrivial solution for Eq. (1.1). Unfortunately, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M9">View MathML</a>, the embedding of E into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M44">View MathML</a> is not compact and Φ no longer satisfies the Palais-Smale condition. Therefore, the ‘standard’ variational methods fail in this case. From this point of view, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M9">View MathML</a> should be seen as a kind of critical exponent for Eq. (1.1). If the potentials V and K are restricted to the class of radially symmetric functions, ‘compactness’ of such a kind is regained and ‘standard’ variational approaches work (see [5] and [6]). However, this method does not seem to apply to the more general equation (1.1) where K and V are non-radially symmetric functions.

It is not easy to deal with Eq. (1.1) directly because there are no known approaches that can be used directly to overcome the difficulty brought by the loss of compactness. However, in this paper, through an interesting transformation, we find an equivalent equation for Eq. (1.1) (see Eq. (2.9) in Section 2). This equation has the advantages that its Palais-Smale sequence can be characterized precisely through the concentration-compactness principle (see Theorem 5.1), and it possesses partial compactness (see Corollary 5.8). By means of these advantages, a positive solution for this equivalent equation and then a corresponding positive solution for Eq. (1.1) are obtained.

Before stating our main result, we need to give some definitions.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M50">View MathML</a>

(1.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M51">View MathML</a>

(1.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M52">View MathML</a>

(1.8)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a> be the Sobolev space endowed with the norm and the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M54">View MathML</a>

respectively, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M55">View MathML</a> be the function space consisting of the functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a> that are p-integrable. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a> can be embedded continuously into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M55">View MathML</a>. Therefore, the infimum

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M60">View MathML</a>

(1.9)

We denote this infimum by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M61">View MathML</a>.

Our main result reads as follows.

Theorem 1.1Under assumptions (A1) and (A2), ifb, sandpsatisfy (1.3) and (1.2) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M62">View MathML</a>

(1.10)

then Eq. (1.1) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M63">View MathML</a>.

Remark 1.2 We should emphasize that condition (1.10) can be satisfied in many situations. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M64">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M65">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M66">View MathML</a> be the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M67">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>. Under assumptions (A1) and (A2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M69">View MathML</a>

Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M72">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M73">View MathML</a>

It follows from this inequality and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M74">View MathML</a> that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M75">View MathML</a> is small enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M76">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M77">View MathML</a>

This implies that (1.10) is satisfied if ϵ is chosen such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M78">View MathML</a>.

Notations Let X be a Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M79">View MathML</a>. We denote the Fréchet derivative of φ at u by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M80">View MathML</a>. The Gateaux derivative of φ is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M82">View MathML</a>. By → we denote the strong and by ⇀ the weak convergence. For a function u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M83">View MathML</a> denotes the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M84">View MathML</a>. The symbol <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M85">View MathML</a> denotes the Kronecker symbol:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M86">View MathML</a>

We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M87">View MathML</a> to mean <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M88">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M89">View MathML</a>.

2 An equivalent equation for Eq. (1.1)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M16">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M91">View MathML</a>. To u, a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M43">View MathML</a> function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a>, we associate a function v, a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M43">View MathML</a> function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M95">View MathML</a>, by the transformation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M96">View MathML</a>

(2.1)

Lemma 2.1Under the above assumptions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M97">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M98">View MathML</a>

(2.3)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M99">View MathML</a>. By direct computations,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M100">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M101">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M102">View MathML</a>

(2.5)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M91">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M106">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M107">View MathML</a>

(2.6)

Substituting (2.6) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M104">View MathML</a> into (2.5) results in

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M109">View MathML</a>

 □

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M110">View MathML</a>

(2.7)

From the classical Hardy inequality (see, e.g., [[13], Lemma 2.1]), we deduce that for every bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M111">View MathML</a> domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M112">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M113">View MathML</a> such that, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M114">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M115">View MathML</a>

(2.8)

Theorem 2.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M116">View MathML</a>is a weak solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M117">View MathML</a>

(2.9)

i.e., for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M118">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M119">View MathML</a>

(2.10)

anduis defined by (2.1), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M114">View MathML</a>and it is a weak solution of (1.1), i.e., for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M121">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M122">View MathML</a>

(2.11)

Proof Using the spherical coordinates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M123">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M126">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M127">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M128">View MathML</a>. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M129">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M130">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M131">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M132">View MathML</a>

(2.12)

Here, we used <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M133">View MathML</a> in the last inequality above. From (2.4), (2.12) and (2.8), we deduce that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37">View MathML</a> such that for every bounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M112">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M136">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M137">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M114">View MathML</a>. Then, to prove that u satisfies (2.11) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M121">View MathML</a>, it suffices to prove that (2.11) holds for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M140">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M140">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M142">View MathML</a> be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M143">View MathML</a>

By using the divergence theorem and Lemma 2.1, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M144">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M145">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M146">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M147">View MathML</a>

This completes the proof. □

This theorem implies that the problem of looking for solutions of (1.1) can be reduced to a problem of looking for solutions of (2.9).

3 The variational functional for Eq. (2.9)

The following inequality is a variant Hardy inequality.

Lemma 3.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M148">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M149">View MathML</a>

(3.1)

Proof We only give the proof of (3.1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M150">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M27">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M150">View MathML</a>, we have the following identity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M154">View MathML</a>

By using the Hölder inequality, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M155">View MathML</a>

Then we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M156">View MathML</a>

 □

From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M157">View MathML</a> (see (2.3)), it is easy to verify that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M159">View MathML</a>

(3.2)

Lemma 3.2There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M160">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M161">View MathML</a>such that for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M163">View MathML</a>

Proof From conditions (A1) and (A2), we deduce that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M165">View MathML</a>

(3.3)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M166">View MathML</a>

by (3.3) and the classical Hardy inequality (see, e.g., [13])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M167">View MathML</a>

we deduce that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M169">View MathML</a>

This together with the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M170">View MathML</a> yields that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M161">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M172">View MathML</a>

(3.4)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M23">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M174">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M175">View MathML</a>

(3.5)

In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M176">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M177">View MathML</a>

(3.6)

Conditions (A1) and (A2) imply that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M179">View MathML</a>

(3.7)

Combining (3.5)-(3.7) yields that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M160">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M181">View MathML</a>

(3.8)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M24">View MathML</a>, (3.7) still holds. From Lemma 3.1 and (3.7), we deduce that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M160">View MathML</a> such that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M185">View MathML</a>

(3.9)

Then the desired result of this lemma follows from (3.4), (3.8) and (3.9) immediately. □

This lemma implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M186">View MathML</a>

(3.10)

is equivalent to the standard norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M187">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>. We denote the inner product associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M189">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M190">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M191">View MathML</a>

(3.11)

By the Sobolev inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M192">View MathML</a>

(3.12)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M193">View MathML</a>

(3.13)

By conditions (A1) and (A2), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M23">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M195">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a>. Therefore, by (3.13), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M198">View MathML</a>

(3.14)

However, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M195">View MathML</a> has a singularity at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M201">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M202">View MathML</a>

(3.15)

Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M204">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M24">View MathML</a>. Then, by the Hardy-Sobolev inequality (see, for example, [[14], Lemma 3.2]), we deduce that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37">View MathML</a> such that (3.14) still holds. Therefore, the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M207">View MathML</a>

(3.16)

is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M43">View MathML</a> functional defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>. Moreover, it is easy to check that the Gateaux derivative of J is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M210">View MathML</a>

and the critical points of J are nonnegative solutions of (2.9).

4 Some minimizing problems

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M211">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M213">View MathML</a>

(4.1)

By this definition, we have, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M215">View MathML</a>

(4.2)

From

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M216">View MathML</a>

we deduce that the norm defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M217">View MathML</a>

(4.3)

is equivalent to the standard norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M187">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>. The inner product corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M220">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M221">View MathML</a>

Lemma 4.1The infimum

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M222">View MathML</a>

(4.4)

is independent of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M223">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212">View MathML</a>.

Proof In this proof, we always view a vector in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a> as a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M226">View MathML</a> matrix, and we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M227">View MathML</a> to denote the conjugate matrix of a matrix A.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M228">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M229">View MathML</a>, let G be an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M230">View MathML</a> orthogonal matrix such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M231">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M16">View MathML</a>. The assumption that G is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M230">View MathML</a> orthogonal matrix implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M236">View MathML</a>, where I is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M230">View MathML</a> identity matrix. Then it is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M238">View MathML</a>

(4.5)

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M239">View MathML</a>

(4.6)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M236">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M241">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M242">View MathML</a>

(4.7)

By (4.6) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M231">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M244">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M245">View MathML</a>

(4.8)

By (4.5), (4.7) and (4.8), we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M246">View MathML</a>. This together with (4.5) leads to the result of this lemma. □

Since the infimum (4.4) is independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M223">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212">View MathML</a>, we denote it by S.

Lemma 4.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M61">View MathML</a>be the infimum in (1.9). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M250">View MathML</a>.

Proof Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M251">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M220">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M253">View MathML</a>

By Lemma 4.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M254">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M255">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M256">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M257">View MathML</a>

 □

Since the functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M258">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M259">View MathML</a> are invariant by translations, the same argument as the proof of [[11], Theorem 1.34] yields that there exists a positive minimizer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M260">View MathML</a> for the infimum S. Moreover, from the Lagrange multiplier rule, it is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M261">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M262">View MathML</a> is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M263">View MathML</a>

(4.9)

In the next section, we shall show that Eq. (4.9) is the ‘limit’ equation of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M264">View MathML</a>

(4.10)

It is easy to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M265">View MathML</a>

(4.11)

is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M43">View MathML</a> functional defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>, the Gateaux derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M268">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M269">View MathML</a>

and the critical points of this functional are solutions of (4.9).

Lemma 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M223">View MathML</a>satisfy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M272">View MathML</a>is a critical point of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M268">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M274">View MathML</a>

(4.12)

Proof Since u is a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M268">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M276">View MathML</a>

(4.13)

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M277">View MathML</a>

(4.14)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M272">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M279">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M280">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M281">View MathML</a>

This together with (4.14) yields the result of this lemma. □

5 The Palais-Smale condition for the functional J

Recall that J is the functional defined by (3.16). By a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282">View MathML</a> sequence of J, we mean a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M283">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M284">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M285">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286">View MathML</a> denotes the dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>. J is called satisfying the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282">View MathML</a> condition if every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282">View MathML</a> sequence of J contains a convergent subsequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>.

Our main result in this section reads as follows.

Theorem 5.1Under assumptions (A1) and (A2), let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M283">View MathML</a>be a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282">View MathML</a>sequence ofJ. Then replacing<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M295">View MathML</a>if necessary by a subsequence, there exist a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M296">View MathML</a>of Eq. (4.10), a finite sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M297">View MathML</a>, kfunctions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M298">View MathML</a>andksequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M299">View MathML</a>satisfying:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M300">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M303">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M304">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M306">View MathML</a>,

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M307">View MathML</a>.

This theorem gives a precise representation of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282">View MathML</a> sequence for the functional J. Through it, partial compactness for J can be regained (see Corollary 5.8).

To prove this theorem, we need some lemmas. Our proof of this theorem is inspired by the proof of [[11], Theorem 8.4].

Lemma 5.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M158">View MathML</a>. Then, for any sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M310">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M311">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M314">View MathML</a>

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M315">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M195">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a>. In this case, the result of this lemma is obvious. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M24">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M319">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M320">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M204">View MathML</a>, by Lemma 3.2 of [14], the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M322">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M323">View MathML</a> is compact. Therefore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M325">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M326">View MathML</a>

And there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M327">View MathML</a> depending only on ϵ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M328">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M329">View MathML</a>. Then, for every n,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M330">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M331">View MathML</a>. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M332">View MathML</a>.

Using the same argument as above, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M334">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M335">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M336">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M337">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M338">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M339">View MathML</a>. Then, using the Lebesgue theorem and the above two inequalities, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M340">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M332">View MathML</a>. Then we get the desired result of this lemma. □

Lemma 5.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M342">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M295">View MathML</a>is bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M345">View MathML</a>

(5.1)

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M346">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M55">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M204">View MathML</a>, by Lemma 3.2 of [14], the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M322">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M323">View MathML</a> is compact. Therefore, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M325">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M353">View MathML</a>

And there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M327">View MathML</a> depending only on ϵ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M328">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M329">View MathML</a>. By (5.1) and the Lions lemma (see, for example, [[11], Lemma 1.21]), we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M357">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M358">View MathML</a>. Now let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M332">View MathML</a>. □

Lemma 5.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M310">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M361">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M363">View MathML</a>

One can follow the proof of [[11], Lemma 8.1] step by step and use Lemma 5.2 to give the proof of this lemma.

The following lemma is a variant Brézis-Lieb lemma (see [15]) and its proof is similar to that of [[11], Lemma 1.32].

Lemma 5.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M283">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M310">View MathML</a>. If

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M295">View MathML</a>is bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>,

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M368">View MathML</a>a.e. on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M370">View MathML</a>

Proof Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M371">View MathML</a>

Then j is a convex function. From [[15], Lemma 3], we have that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M373">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M374">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M375">View MathML</a>

(5.2)

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M376">View MathML</a>

By Lemma 3.2 of [14], the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M322">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M323">View MathML</a> is compact. We get that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M325">View MathML</a> such that for any n,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M380">View MathML</a>

(5.3)

And there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M327">View MathML</a> depending only on ϵ such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M328">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M329">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M384">View MathML</a>

By the Lebesgue theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M385">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>. This together with (5.3) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M387">View MathML</a>

The left proof is the same as the proof of [[11], Lemma 1.32]. □

Lemma 5.6If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M388">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M389">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M391">View MathML</a>is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M392">View MathML</a>

Proof (1) Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M361">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>, we get that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M396">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M397">View MathML</a>

(5.4)

(2) Lemma 5.5 implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M398">View MathML</a>

(5.5)

By (5.4), (5.5) and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M284">View MathML</a>, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M400">View MathML</a>

(3) Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M285">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M361">View MathML</a>, it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M389">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M405">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M406">View MathML</a>

(5.6)

By Lemma 5.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M407">View MathML</a>

(5.7)

Combining (5.6) and (5.7) leads to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M408">View MathML</a>. Then, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M285">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M389">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M412">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286">View MathML</a>. □

Lemma 5.7If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M312">View MathML</a>and as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M416">View MathML</a>

then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M223">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M419">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M420">View MathML</a>is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M421">View MathML</a>

Proof We divide the proof into several steps.

(1) Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M422">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>, it is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M424">View MathML</a>

(2) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M405">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M426">View MathML</a>

(5.8)

By the definition of the inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M190">View MathML</a> (see (3.11)), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M428">View MathML</a>

(5.9)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M422">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M431">View MathML</a>

(5.10)

By assumption (A2) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M432">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M433">View MathML</a>. This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M434">View MathML</a>

Moreover, together with (2.8) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M312">View MathML</a> yields that for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M436">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M437">View MathML</a>

Combining the above two limits leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M438">View MathML</a>

(5.11)

By (5.11) and the Hölder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M439">View MathML</a>

(5.12)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M440">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M70">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M442">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M443">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M444">View MathML</a>

(5.13)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M445">View MathML</a>

(5.14)

where the constant C is independent of ϵ and n. There exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M446">View MathML</a>, denoted by itself for convenience, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M223">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M212">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M449">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>. Then, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M312">View MathML</a>, we get that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M453">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M454">View MathML</a> converges to θ uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M455">View MathML</a>. Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M456">View MathML</a> such that, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M457">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M458">View MathML</a>

(5.15)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M422">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M461">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M462">View MathML</a>. It implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M463">View MathML</a>

This together with (5.14), (5.15) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M464">View MathML</a>

yields that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M465">View MathML</a> such that, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M466">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M467">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M468">View MathML</a>

(5.16)

Combining (5.10), (5.12) and (5.16) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M469">View MathML</a>

(5.17)

We obtain, by the Hölder inequality and Lemma 5.2, that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M471">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M472">View MathML</a> and C are positive constants independent of n and h. This together with (5.8) and (5.17) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M473">View MathML</a>

(5.18)

Then, by the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M285">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M476">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M477">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M419">View MathML</a>.

(3) From the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M479">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M480">View MathML</a>

(5.19)

By the definition of the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M189">View MathML</a> (see (3.10)), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M482">View MathML</a>

(5.20)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M483">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M484">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a>, using the Lebesgue convergence theorem, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M486">View MathML</a>

(5.21)

By (5.11), (5.20) and (5.21), we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M487">View MathML</a>

(5.22)

Combining (5.19), (5.22) and (5.17) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M488">View MathML</a>

(5.23)

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M489">View MathML</a>

(5.24)

We obtain from Lemma 5.5 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M490">View MathML</a>

(5.25)

By Lemma 5.2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M491">View MathML</a>

(5.26)

Combining (5.24)-(5.26) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M492">View MathML</a>

(5.27)

Combining (5.23), (5.27) and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M284">View MathML</a> leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M494">View MathML</a>

(4) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M405">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M496">View MathML</a>

(5.28)

We shall give the limits for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M497">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M498">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>.

First, as (5.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M500">View MathML</a>

By the Hölder inequality and (5.11), we get that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M502">View MathML</a>

Thus, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M504">View MathML</a>

holds uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501">View MathML</a>. Moreover, a similar argument as the proof of (5.16) yields that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M507">View MathML</a>

holds uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501">View MathML</a>. Therefore, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M510">View MathML</a>

(5.29)

holds uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501">View MathML</a>.

Second, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M422">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a> and Lemma 5.4, we deduce that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M515">View MathML</a>

(5.30)

holds uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501">View MathML</a>. By the Hölder inequality, (3.14) and Lemma 5.2, we get that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M518">View MathML</a>

(5.31)

By Lemma 5.2, we get that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M436">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M521">View MathML</a>

(5.32)

Combining (5.31) and (5.32) yields that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M522">View MathML</a>

(5.33)

holds uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501">View MathML</a>. Then, by (5.30), (5.33) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M524">View MathML</a>

we get that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M287">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M526">View MathML</a>

(5.34)

holds uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501">View MathML</a>.

Finally, combining (5.28), (5.29) and (5.34) leads to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M528">View MathML</a>

holds uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M501">View MathML</a>. This together with the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M419">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M285">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286">View MathML</a> yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M412">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M286">View MathML</a>. □

Proof of Theorem 5.1 We divide the proof into two steps.

(1) For n big enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M535">View MathML</a>

(5.35)

As mentioned in Section 3, the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M189">View MathML</a> is equivalent to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M187">View MathML</a>. Therefore, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M37">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M539">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M540">View MathML</a>. Then by (5.35) there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M38">View MathML</a> such that for n big enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M542">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M543">View MathML</a> is bounded.

(2) Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M544">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M546">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a>. By Lemma 5.6, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M548">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M549">View MathML</a> is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M550">View MathML</a>

(5.36)

Let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M551">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M552">View MathML</a>, Lemma 5.3 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M553">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M55">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M555">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M557">View MathML</a>

and the proof is complete. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M558">View MathML</a>, we may assume the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M559">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M560">View MathML</a>

Let us define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M561">View MathML</a>. We may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M562">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M564">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M25">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M566">View MathML</a>

it follows from the Rellich theorem that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M567">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M568">View MathML</a>. But <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M569">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M53">View MathML</a>, so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M571">View MathML</a> is unbounded. We may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M572">View MathML</a>. Finally, by (5.36) and Lemma 5.7, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M573">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M574">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M575">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M576">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M577">View MathML</a>

Moreover, Lemma 4.3 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M578">View MathML</a>

Iterating the above procedure, we construct sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M579">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M580">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M581">View MathML</a>. Since for every l, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M582">View MathML</a>, the iteration must terminate at some finite index k. This finishes the proof of this theorem. □

The following corollary is a direct consequence of Theorem 5.1 and Lemma 4.3. It implies that the functional J satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M282">View MathML</a> condition if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M584">View MathML</a>.

Corollary 5.8Under assumptions (A1) and (A2), any sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M283">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M586">View MathML</a>

contains a convergent subsequence.

6 Proof of Theorem 1.1

Recall that the critical points of J are nonnegative solutions of (2.9). By Corollary 2.2, to prove that Eq. (1.1) has a positive solution, it suffices to prove that J has a nontrivial critical point. Moreover, by Corollary 5.8, it suffices to apply the classical mountain pass theorem (see, e.g., [[11], Theorem 1.15]) to J with the mountain pass value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M584">View MathML</a>.

By assumption (1.10) and Lemma 4.2, there exists a nonnegative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M588">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M589">View MathML</a>

We obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M590">View MathML</a>

(6.1)

By (3.14), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M591">View MathML</a>

Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M64">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M593">View MathML</a>

Moreover, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M594">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M595">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M596">View MathML</a>. It follows from (6.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M597">View MathML</a>

By Corollary 5.8 and the mountain pass theorem (see [[11], Theorem 1.15]), J has a critical value c such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M598">View MathML</a> and Eq. (2.9) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M148">View MathML</a>. Then, by Theorem 2.2, the function u defined by (2.1) is a positive solution of (1.1). To complete the proof, it suffices to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M63">View MathML</a>. Using the divergence theorem, Lemma 2.1 and (2.12), we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M601">View MathML</a>

Moreover, by Lemma 2.1 and (2.12), we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M602">View MathML</a>

(6.2)

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/201/mathml/M603">View MathML</a>.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Shaowei Chen was supported by Science Foundation of Huaqiao University.

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