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Existence and multiplicity of positive solutions to a perturbed singular elliptic system deriving from a strongly coupled critical potential

Tsing-San Hsu

Author Affiliations

Department of Natural Sciences in the Center for General Education, Chang Gung University, Kwei-San, Tao-Yuan, 333, Taiwan, ROC

Boundary Value Problems 2012, 2012:116  doi:10.1186/1687-2770-2012-116


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/116


Received:12 March 2012
Accepted:3 October 2012
Published:17 October 2012

© 2012 Hsu; 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 consider singular elliptic systems involving a strongly coupled critical potential and concave nonlinearities. By using variational methods and analytical techniques, the existence and multiplicity of positive solutions to the system are established.

MSC: 35J60, 35B33.

Keywords:
Palais-Smale condition; Nehari manifold; strongly coupled; elliptic system; critical potential

1 Introduction and main results

In this paper, we consider the following elliptic system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M2">View MathML</a> is a smooth bounded domain such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M5">View MathML</a> is the critical Sobolev exponent, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M6">View MathML</a> is the best Hardy constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M7">View MathML</a> denotes the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M8">View MathML</a> with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M10">View MathML</a> is defined as the completion of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M8">View MathML</a> with respect to the norm defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M12">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M13">View MathML</a>.

Definitions of strongly and weakly coupled terms are as follows.

The terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M15">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M16">View MathML</a>) are weakly coupled, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M17">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M16">View MathML</a>) is strongly coupled when L or K is a derivative operator. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M19">View MathML</a> is strongly coupled when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M21">View MathML</a> are positive.

The parameters in (1.1) satisfy the following assumption.

(ℋ) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M30">View MathML</a>.

The corresponding energy functional of (1.1) is defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M31">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M32">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M33">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M34">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M35">View MathML</a> and the duality product between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M31">View MathML</a> and its dual space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M37">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M38">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M39">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M40">View MathML</a> denotes the Fréchet derivative of J at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M41">View MathML</a>. A pair of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M42">View MathML</a> is said to be a weak solution of (1.1) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M43">View MathML</a>

Therefore, a weak solution of (1.1) is equivalent to a nonzero critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M44">View MathML</a>[1].

Problem (1.1) is related to the well-known Hardy inequality [2]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M45">View MathML</a>

(1.2)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M13">View MathML</a>, by (1.2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M47">View MathML</a> is an equivalent norm of H, the operator L is positive and the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M48">View MathML</a> of L and the following best constant are well defined:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M49">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M50">View MathML</a> is the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M51">View MathML</a> with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M52">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M53">View MathML</a> is the well-known best Sobolev constant. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M23">View MathML</a>, the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M55">View MathML</a> is achieved by the following extremal functions [3]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M56">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M57">View MathML</a> is a radially symmetric function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M58">View MathML</a>

On the other hand, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M30">View MathML</a>, by the Young and Sobolev inequalities, the following best constants are well defined on the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M65">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M66">View MathML</a>

(1.5)

We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M67">View MathML</a>

(1.6)

Since f is a continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M68">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M69">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M70">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M71">View MathML</a>

(1.7)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M75">View MathML</a>. Then (1.1) reduces to the semilinear scalar problems that have been extensively investigated by many authors. See [4-6] and the references therein.

Regular semilinear elliptic systems have been studied extensively and many conclusions have been established. For example, Alves et al. studied in [7] an elliptic system and some important conclusions had been obtained. However, the elliptic systems involving the Hardy inequality have seldom been studied and we only find some results in [8-16]. Thus it is necessary for us to investigate the related singular systems deeply. Among the references above, the elliptic systems involving the Hardy inequality and concave-convex nonlinearities had been studied only in [12]. In this paper, only the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M24">View MathML</a> of (1.1) involving multiple strongly-coupled critical terms is considered.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M77">View MathML</a> be the Lebesgue measure of Ω. We define the following constant:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M78">View MathML</a>

(1.8)

Then the main results of this paper can be concluded in the following theorems and the conclusions are new to the best of our knowledge. It can be verified that the intervals in Theorems 1.1 and 1.2 for the parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M80">View MathML</a>, μ and q are allowable.

Theorem 1.1Suppose that (ℋ) holds and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M81">View MathML</a>. Then problem (1.1) has at least one positive solution.

Theorem 1.2Suppose that (ℋ) holds, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M83">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M84">View MathML</a>. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M85">View MathML</a>such that problem (1.1) has at least two positive solutions for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M79">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M80">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M88">View MathML</a>.

This paper is organized as follows. Some preliminary results and properties of the Nehari manifold are established in Sections 2 and 3, and Theorems 1.1 and 1.2 are proved in Section 4.

2 The local Palais-Smale condition

Throughout this paper, we always assume that the assumption (ℋ) holds, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M89">View MathML</a> denotes the norm of the space H, by the Hardy inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M90">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M91">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M92">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M93">View MathML</a> denotes the first eigenvalue of the operator L, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M94">View MathML</a> means the norm of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M95">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M96">View MathML</a> is the dual space of E. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M97">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M99">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M100">View MathML</a> is said to be nonnegative in Ω if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M102">View MathML</a> in Ω. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M100">View MathML</a> is said to be positive in Ω if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M105">View MathML</a> in Ω. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M106">View MathML</a> is a ball in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M107">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M108">View MathML</a> denotes a quantity satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M110">View MathML</a> means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M111">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M113">View MathML</a> is a generic infinitesimal value. In particular, the quantity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M114">View MathML</a> means that there exist the constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M115">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M116">View MathML</a> as ε is small. We always denote positive constants as C and omit dx in integrals for convenience.

Lemma 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M117">View MathML</a>is a (PS)c-sequence ofJwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M118">View MathML</a>inE, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M119">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M120">View MathML</a>, where

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M100">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124">View MathML</a> is a (PS)c-sequence of J with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M118">View MathML</a> in E, we can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M119">View MathML</a>, and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M127">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M128">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M129">View MathML</a>

From the Hölder inequality it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M130">View MathML</a>

Thus, the proof is complete. □

Lemma 2.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M117">View MathML</a>is a (PS)c-sequence of the functionalJ, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124">View MathML</a>is bounded inE.

Proof See Hsu [[12], Lemma 2.2]. □

Lemma 2.3Suppose that (ℋ) holds. ThenJsatisfies the (PS)ccondition for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M133">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M134">View MathML</a>

(2.1)

Proof We follow the argument in [15]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M135">View MathML</a> be a (PS)c-sequence of J with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M133">View MathML</a>. Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M122">View MathML</a>. We know from Lemma 2.2 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124">View MathML</a> is bounded in E, and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M139">View MathML</a> up to a subsequence, z is a critical point of J. Furthermore, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M141">View MathML</a> weakly in H and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M143">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M144">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M145">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M143">View MathML</a> a.e. in Ω. Hence, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M148">View MathML</a>

(2.2)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M150">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M151">View MathML</a>. From the Brézis-Lieb lemma [17] it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M152">View MathML</a>

(2.3)

and by Lemma 2.1 in [18] we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M153">View MathML</a>

(2.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M155">View MathML</a> and by (2.2) to (2.4), we can deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M156">View MathML</a>

(2.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M157">View MathML</a>

Hence, we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M158">View MathML</a>

(2.6)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M159">View MathML</a>, the proof is complete. Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M160">View MathML</a>; then from (2.6) and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M161">View MathML</a> it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M162">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M163">View MathML</a>

(2.7)

In addition, from (2.5) to (2.7) and Lemma 2.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M164">View MathML</a>

which is a contradiction. Therefore, the proof of Lemma 2.3 is complete. □

3 Nehari manifold

Since J is unbounded below on E, we need to consider J on the Nehari manifold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M165">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M166">View MathML</a> if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M167">View MathML</a>

(3.1)

By the Hölder inequality and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M93">View MathML</a> it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M169">View MathML</a>

(3.2)

Lemma 3.1The functionalJis coercive and bounded below on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170">View MathML</a>.

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M171">View MathML</a>. From (3.1) and (3.2) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M172">View MathML</a>

(3.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M173">View MathML</a>

(3.4)

Thus, J is coercive and bounded below on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170">View MathML</a>. □

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M175">View MathML</a>. Then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M176">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M177">View MathML</a>

(3.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M178">View MathML</a>

(3.6)

We split <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170">View MathML</a> into three parts:

Lemma 3.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M98">View MathML</a>is a local minimizer ofJon<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M183">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M119">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M96">View MathML</a>.

Proof The proof is similar to that of [19] and the details are omitted. □

Lemma 3.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M186">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M187">View MathML</a>.

Proof We argue by contradiction. Suppose that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M188">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M190">View MathML</a>. Then the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M191">View MathML</a> together with (3.5) and (3.6) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M192">View MathML</a>

(3.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M193">View MathML</a>

(3.8)

By (1.5) and (3.7) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M194">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M195">View MathML</a>

(3.9)

By (3.2) and (3.8) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M196">View MathML</a>

(3.10)

From (3.9) and (3.10) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M197">View MathML</a>

which is a contradiction. □

By Lemma 3.3, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M198">View MathML</a> and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M199">View MathML</a>

Lemma 3.4

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M200">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M201">View MathML</a>.

(ii) There exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M202">View MathML</a>depending on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M80">View MathML</a>, q, N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M93">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M77">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M208">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M209">View MathML</a>.

Proof (i) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M210">View MathML</a>. By (3.1) and (3.6) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M211">View MathML</a>

(3.11)

According to (3.1) and (3.11), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M212">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M200">View MathML</a>.

(ii) Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M214">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M215">View MathML</a>. By (1.7), (3.1) and (3.5) we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M216">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M217">View MathML</a>

(3.12)

From (3.4) and (3.12) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M218">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M219">View MathML</a> is a positive constant. □

Lemma 3.5Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M201">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M98">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M222">View MathML</a>. Then there exist unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M223">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M224">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M225">View MathML</a>. In particular, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M226">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M227">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M228">View MathML</a>.

Proof The proof is similar to that of [20] and is omitted. □

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M98">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M230">View MathML</a>, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M231">View MathML</a>

Then we have the following lemma.

Lemma 3.6Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M201">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M233">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M230">View MathML</a>. Then there exist unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M235">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M236">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M225">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M238">View MathML</a>

Proof The proof is almost the same as that in [[20], Lemma 2.7] and is omitted here. □

4 Proof of Theorems 1.1 and 1.2

Lemma 4.1

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M201">View MathML</a>, then the functionalJhas a (PS)-sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M241">View MathML</a>.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M242">View MathML</a>, then the functionalJhas a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M243">View MathML</a>-sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M244">View MathML</a>.

Proof The proof is similar to that of [21] and is omitted. □

Lemma 4.2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M187">View MathML</a>. ThenJhas a minimizer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M246">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247">View MathML</a>is a positive solution of (1.1) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M248">View MathML</a>.

Proof By Lemma 4.1(i), there exists a (PS)-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M241">View MathML</a> of J such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M251">View MathML</a>

(4.1)

Since J is coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170">View MathML</a> (see Lemma 3.1), we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124">View MathML</a> is bounded in E. Passing to a subsequence (still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124">View MathML</a>), we can assume that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M255">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M256">View MathML</a>

(4.2)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M257">View MathML</a>

(4.3)

First, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247">View MathML</a> is a solution of (1.1). By (4.1) and (4.2), it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247">View MathML</a> is a solution of (1.1). Furthermore, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M241">View MathML</a> and (3.3), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M261">View MathML</a>

(4.4)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M262">View MathML</a> in (4.4), by (4.1), (4.2) and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M263">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M264">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M265">View MathML</a> is a nontrivial solution of (1.1).

Next, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M266">View MathML</a> strongly in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M267">View MathML</a>. Noting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M265">View MathML</a> and applying the Fatou lemma, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M269">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M270">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M271">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M272">View MathML</a>. By the Brézis-Lieb lemma [17], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M273">View MathML</a>

Then standard argument shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M266">View MathML</a> strongly in E. Moreover, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M246">View MathML</a>. Otherwise, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M276">View MathML</a>, then by Lemma 3.5 there exist unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M277">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M278">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M279">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M280">View MathML</a>

there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M281">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M282">View MathML</a>. By Lemma 3.5 we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M283">View MathML</a>

which is a contradiction. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M284">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M285">View MathML</a>, by Lemma 3.2 we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247">View MathML</a> is a nontrivial nonnegative solution of (1.1).

In particular <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M288">View MathML</a>. Indeed, without loss of generality, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M289">View MathML</a>. Then as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M290">View MathML</a> is a nontrivial nonnegative solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M291">View MathML</a>

by the standard regularity theory, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M292">View MathML</a> in Ω and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M293">View MathML</a>

Moreover, we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M294">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M295">View MathML</a>

Now,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M296">View MathML</a>

and so by Lemma 3.6 there is unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M297">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M298">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M299">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M300">View MathML</a>

This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M301">View MathML</a>

which is a contradiction.

Finally, from the maximum principle [22] we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M302">View MathML</a> in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247">View MathML</a> is thus a positive solution of (1.1). □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M304">View MathML</a> be defined as in (1.4) and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M305">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M306">View MathML</a> is a cut-off function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M307">View MathML</a>

The following results are already known.

Lemma 4.3[4]

As<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M112">View MathML</a>we have the following estimates:

(4.5)

(4.6)

(4.7)

Lemma 4.4[11]

Suppose that (ℋ) holds, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M312">View MathML</a>is defined as in (1.6) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M304">View MathML</a>are the minimizers of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M55">View MathML</a>defined as in (1.4). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M315">View MathML</a>and has the minimizers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M316">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M317">View MathML</a>.

Lemma 4.5Under the assumptions of Theorem 1.2, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M318">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M319">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M320">View MathML</a>there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M321">View MathML</a>

(4.8)

In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M322">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M320">View MathML</a>.

Proof For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M324">View MathML</a>, define the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M325">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M326">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M327">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M328">View MathML</a> as t is closed to 0. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M329">View MathML</a> is attained at some finite <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M330">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M331">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M332">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M333">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M334">View MathML</a> are the positive constants independent of ε.

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M335">View MathML</a> small enough such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M336">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M337">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M338">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M339">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M324">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M188">View MathML</a>, which implies that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M342">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M343">View MathML</a>, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M337">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M345">View MathML</a>

(4.9)

From (4.9) and Lemmas 4.3, 4.4 it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M346">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M347">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M348">View MathML</a>

where we have used the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M349">View MathML</a>.

Therefore we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M350">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M351">View MathML</a> such that

The definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M353">View MathML</a> in Lemma 2.1 implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M354">View MathML</a>

Note that

Taking ε small enough, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M356">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M357">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M358">View MathML</a>

(4.10)

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M359">View MathML</a>. Then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M360">View MathML</a> there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M361">View MathML</a>

(4.11)

Finally, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M362">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M320">View MathML</a>. Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M338">View MathML</a>. By Lemma 3.5, the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M365">View MathML</a> and (4.11), we can deduce that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M366">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M367">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M368">View MathML</a>

The proof is thus complete by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M369">View MathML</a>. □

Lemma 4.6Set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M370">View MathML</a>. Then for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M371">View MathML</a>, problem (1.1) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M372">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M373">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M374">View MathML</a>.

Proof By Lemma 4.1, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M375">View MathML</a>-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M376">View MathML</a> of J for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M377">View MathML</a>. From Lemmas 2.3, 3.4 and 4.5, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M378">View MathML</a> and J satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M379">View MathML</a> condition for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M88">View MathML</a>. Since J is coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M170">View MathML</a>, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124">View MathML</a> is bounded in E. Therefore, there exist a subsequence (still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M124">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M384">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M385">View MathML</a> strongly in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M386">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M88">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M388">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M389">View MathML</a>, by Lemma 3.2 we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M372">View MathML</a> is a nontrivial nonnegative solution of (1.1). Moreover, by (3.7) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M391">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M392">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M393">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M394">View MathML</a>. From the strong maximum principle [22] it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M372">View MathML</a> is a positive solution of (1.1). □

Proof of Theorems 1.1 and 1.2. By Lemma 4.2, we obtain that (1.1) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M396">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M81">View MathML</a>. On the other hand, from Lemma 4.6, we can get the second positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M373">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M399">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M400">View MathML</a>, this implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M247">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/116/mathml/M372">View MathML</a> are distinct. □

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author was grateful for the referee’s helpful suggestions and comments.

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