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Ground state homoclinic orbits of damped vibration problems

Guan-Wei Chen* and Jian Wang

Author Affiliations

School of Mathematics and Statistics, Anyang Normal University, Anyang, Henan, 455000, P.R. China

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Boundary Value Problems 2014, 2014:106  doi:10.1186/1687-2770-2014-106

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/106


Received:6 January 2014
Accepted:24 April 2014
Published:9 May 2014

© 2014 Chen and Wang; 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 credited.

Abstract

In this paper, we consider a class of non-periodic damped vibration problems with superquadratic nonlinearities. We study the existence of nontrivial ground state homoclinic orbits for this class of damped vibration problems under some conditions weaker than those previously assumed. To the best of our knowledge, there has been no work focused on this case.

MSC: 49J40, 70H05.

Keywords:
non-periodic damped vibration problems; ground state homoclinic orbits; superquadratic nonlinearity

1 Introduction and main results

We shall study the existence of ground state homoclinic orbits for the following non-periodic damped vibration system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M1">View MathML</a>

(1.1)

where M is an antisymmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M2">View MathML</a> constant matrix, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M3">View MathML</a> is a symmetric matrix, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M5">View MathML</a> denotes its gradient with respect to the u variable. We say that a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M6">View MathML</a> of (1.1) is homoclinic (to 0) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M7">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M9">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M10">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M11">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M6">View MathML</a> is called a nontrivial homoclinic solution.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M13">View MathML</a> (zero matrix), then (1.1) reduces to the following second-order Hamiltonian system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M14">View MathML</a>

(1.2)

This is a classical equation which can describe many mechanic systems such as a pendulum. In the past decades, the existence and multiplicity of periodic solutions and homoclinic orbits for (1.2) have been studied by many authors via variational methods; see [1-18] and the references therein.

The periodic assumptions are very important in the study of homoclinic orbits for (1.2) since periodicity is used to control the lack of compactness due to the fact that (1.2) is set on all ℝ. However, non-periodic problems are quite different from the ones described in periodic cases. Rabinowitz and Tanaka [11] introduced a type of coercive condition on the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M15">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M16">View MathML</a>

(1.3)

and obtained the existence of a homoclinic orbit for non-periodic (1.2) under the Ambrosetti-Rabinowitz (AR) superquadratic condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M17">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M18">View MathML</a> is a constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M19">View MathML</a> denotes the standard inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M20">View MathML</a> and the associated norm is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M21">View MathML</a>.

We should mention that in the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M22">View MathML</a>, i.e., the damped vibration system (1.1), only a few authors have studied homoclinic orbits of (1.1); see [19-23]. Zhu [23] considered the periodic case of (1.1) (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M24">View MathML</a> are T-periodic in t with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M25">View MathML</a>) and obtained the existence of nontrivial homoclinic solutions of (1.1). The authors [19-22] considered the non-periodic case of (1.1): Zhang and Yuan [22] obtained the existence of at least one homoclinic orbit for (1.1) when H satisfies the subquadratic condition at infinity by using a standard minimizing argument; by a symmetric mountain pass theorem and a generalized mountain pass theorem, Wu and Zhang [21] obtained the existence and multiplicity of homoclinic orbits for (1.1) when H satisfies the local (AR) superquadratic growth condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M26">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M28">View MathML</a> are two constants. Notice that the authors [21,22] all used condition (1.3). Recently, the author in [19,20] obtained infinitely many homoclinic orbits for (1.1) when H satisfies the subquadratic[19] and asymptotically quadratic[20] condition at infinity by the following weaker conditions than (1.3):

(L1) There is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M29">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M30">View MathML</a>

(L2) There is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M31">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M32">View MathML</a>

which were firstly used in [15]. It is not hard to check that the matrix-valued function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M33">View MathML</a> satisfies (L1) and (L2), but does not satisfy (1.3).

We define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M34">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M35">View MathML</a>

Since M is an antisymmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M2">View MathML</a> constant matrix, Γ is self-adjoint on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M37">View MathML</a>. Let χ denote the self-adjoint extension of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M38">View MathML</a>. We are interested in the indefinite case:

(J1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M39">View MathML</a>.

To state our main result, we still need the following assumptions:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M40">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M44">View MathML</a>.

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M45">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M44">View MathML</a>.

(H3) For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M49">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M50">View MathML</a>

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M51">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M52">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M54">View MathML</a>

(1.5)

(H5) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M56">View MathML</a>, there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M57">View MathML</a>

Our main results read as follows.

Theorem 1.1If (L1)-(L2), (J1) and (H1)-(H5) hold, then (1.1) has at least one nontrivial homoclinic orbit.

Theorem 1.2Letbe the collection of solutions of (1.1), then there is a solution that minimizes the energy functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M58">View MathML</a>

over ℳ, where the spaceEis defined in Section 2. In addition, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M59">View MathML</a>

uniformly int, then there is a nontrivial homoclinic orbit that minimizes the energy functional over<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M60">View MathML</a>, i.e., a ground state homoclinic orbit.

Remark 1.1 Although the authors [21] have studied (1.1) with superquadratic nonlinearities, our superquadratic condition (H4) is weaker than (1.4) in [21]. Moreover, we study the ground state homoclinic orbit of (1.1). To the best of our knowledge, there has been no result published concerning the ground state homoclinic orbit of (1.1).

Example 1.1

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M61">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M62">View MathML</a>,

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M64">View MathML</a> is continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M65">View MathML</a>. It is easy to check that the above two functions satisfy assumptions (H1)-(H5) if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M66">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M67">View MathML</a> is the function in (H4)-(H5).

The rest of the present paper is organized as follows. In Section 2, we establish the variational framework associated with (1.1), and we also give some preliminary lemmas, which are useful in the proofs of our main results. In Section 3, we give the detailed proofs of our main results.

2 Preliminary lemmas

In the following, we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M68">View MathML</a> to denote the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M69">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M70">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M71">View MathML</a> be a Hilbert space with the inner product and norm given respectively by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M72">View MathML</a>

It is well known that W is continuously embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M69">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M74">View MathML</a>. We define an operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M75">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M76">View MathML</a>

Since M is an antisymmetric <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M2">View MathML</a> constant matrix, Γ is self-adjoint on W. Moreover, we denote by χ the self-adjoint extension of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M38">View MathML</a> with the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M79">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M80">View MathML</a>, the domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M81">View MathML</a>. We define respectively on E the inner product and the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M82">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M83">View MathML</a> denotes the inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M84">View MathML</a>.

By a similar proof of Lemma 3.1 in [15], we can prove that if conditions (L1) and (L2) hold, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M85">View MathML</a>

(2.1)

Therefore, it is easy to prove that the spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M86">View MathML</a> has a sequence of eigenvalues (counted with their multiplicities)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M87">View MathML</a>

and the corresponding system of eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M88">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M89">View MathML</a>) forms an orthogonal basis in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M84">View MathML</a>.

By (J1), we may let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M91">View MathML</a>

Then one has the orthogonal decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M92">View MathML</a>

with respect to the inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M93">View MathML</a>. Now, we introduce respectively on E the following new inner product and norm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M94">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M95">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M97">View MathML</a>. Clearly, the norms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M99">View MathML</a> are equivalent (see [4]), and the decomposition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M100">View MathML</a> is also orthogonal with respect to both inner products <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M83">View MathML</a>. Hence, by (J1), E with equivalent norms, besides, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M103">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M104">View MathML</a>

(2.4)

where a and b are defined in (J1).

For problem (1.1), we consider the following functional:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M105">View MathML</a>

Then I can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M106">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M107">View MathML</a>. In view of the assumptions of H, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M108">View MathML</a> and the derivatives are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M109">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M110">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M97">View MathML</a>. By the discussion of [24], the (weak) solutions of system (1.1) are the critical points of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M113">View MathML</a> functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M114">View MathML</a>. Moreover, it is easy to verify that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M115">View MathML</a> is a solution of (1.1), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M117">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M118">View MathML</a> (see Lemma 3.1 in [25]).

The following abstract critical point theorem plays an important role in proving our main result. Let E be a Hilbert space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M98">View MathML</a> and have an orthogonal decomposition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M121">View MathML</a> is a closed and separable subspace. There exists a norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M122">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M123">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M124">View MathML</a> and inducing a topology equivalent to the weak topology of N on a bounded subset of N. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M125">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M127">View MathML</a>, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M128">View MathML</a>. Particularly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M129">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M98">View MathML</a>-bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M131">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M132">View MathML</a> weakly in N, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M133">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M135">View MathML</a> weakly in E (cf.[26]).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M137">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M138">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M139">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M140">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M141">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M142">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M144">View MathML</a>. We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M145">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M146">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M147">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M148">View MathML</a> denotes various finite-dimensional subspaces of E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M149">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M150">View MathML</a>.

The variant weak linking theorem is as follows.

Lemma 2.1 ([26])

The family of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M113">View MathML</a>-functionals<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M152">View MathML</a>has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M153">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M154">View MathML</a>. Assume that

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M157">View MathML</a>;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M158">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M159">View MathML</a>;

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M160">View MathML</a>is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M161">View MathML</a>-upper semicontinuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M162">View MathML</a>is weakly sequentially continuous onE. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M160">View MathML</a>maps bounded sets to bounded sets;

(d) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M164">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M165">View MathML</a>.

Then, for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M166">View MathML</a>, there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M168">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M169">View MathML</a>.

In order to apply Lemma 2.1, we shall prove a few lemmas. We pick <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M170">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M171">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M172">View MathML</a>, we consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M173">View MathML</a>

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M160">View MathML</a> satisfies condition (a) in Lemma 2.1. To see (c), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M176">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M178">View MathML</a> in E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M179">View MathML</a> a.e. on ℝ, going to a subsequence if necessary. It follows from the weak lower semicontinuity of the norm, Fatou’s lemma and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M180">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M182">View MathML</a> by (1.5) in (H4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M183">View MathML</a>

Thus we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M184">View MathML</a>. It implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M160">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M161">View MathML</a>-upper semicontinuous. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M162">View MathML</a> is weakly sequentially continuous on E due to [27].

Lemma 2.2Under assumptions of Theorem 1.1, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M188">View MathML</a>

Proof By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M189">View MathML</a> and (H4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M190">View MathML</a>

which is due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53">View MathML</a>. □

Therefore, Lemma 2.2 implies that condition (b) holds. To continue the discussion, we still need to verify condition (d), that is, the following two lemmas.

Lemma 2.3Under assumptions of Theorem 1.1, there are two positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M192">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M193">View MathML</a>

Proof By (H1), (H3), (2.4) and the Sobolev embedding theorem, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M194">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M195">View MathML</a>

where C is a positive constant. It implies the conclusion if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M196">View MathML</a> sufficiently small. □

Lemma 2.4Under assumptions of Theorem 1.1, then there is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M141">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M198">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M199">View MathML</a>.

Proof Suppose by contradiction that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M200">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M201">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M202">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M203">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M204">View MathML</a>, then by (H2) and (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M205">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M206">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M207">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M208">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M209">View MathML</a>

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M203">View MathML</a> and the definition of I that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M211">View MathML</a>

(2.5)

There are renamed subsequences such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M213">View MathML</a>, and there is a renamed subsequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M214">View MathML</a> in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M215">View MathML</a> a.e. on ℝ.

We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M216">View MathML</a>

(2.6)

Case 1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M217">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M218">View MathML</a> be the subset of ℝ where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M219">View MathML</a>, then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M220">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M221">View MathML</a>. It follows from (H4) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M223">View MathML</a>

Case 2. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M224">View MathML</a>, then by (H4) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M226">View MathML</a>

Therefore, Cases 1 and 2 imply that (2.6) holds. Therefore, by (2.5), (2.6) and the facts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M213">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M229">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M230">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M219">View MathML</a>. It follows from (H4) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M232">View MathML</a>

which contradicts (2.5). The proof is finished. □

Therefore, Lemmas 2.3 and 2.4 imply that condition (d) of Lemma 2.1 holds. Applying Lemma 2.1, we soon obtain the following fact.

Lemma 2.5Under assumptions of Theorem 1.1, for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M166">View MathML</a>, there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M235">View MathML</a>

where the definition of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M236">View MathML</a>is given in Lemma 2.1.

Lemma 2.6Under assumptions of Theorem 1.1, for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M166">View MathML</a>, there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M238">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M239">View MathML</a>

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167">View MathML</a> be the sequence obtained in Lemma 2.5. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167">View MathML</a> is bounded, we can assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M242">View MathML</a> in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M243">View MathML</a> a.e. on ℝ. By (H1), (H3), (2.1) and Theorem A.4 in [27], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M244">View MathML</a>

(2.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M245">View MathML</a>

(2.8)

By Lemma 2.5 and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M162">View MathML</a> is weakly sequentially continuous, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M247">View MathML</a>

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M248">View MathML</a>. By Lemma 2.5, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M249">View MathML</a>

It follows from (2.7), (2.8) and the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M250">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M251">View MathML</a>

The proof is finished. □

Applying Lemma 2.6, we soon obtain the following fact.

Lemma 2.7Under assumptions of Theorem 1.1, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M166">View MathML</a>, there are sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M253">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M201">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M213">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M256">View MathML</a>

Lemma 2.8Under assumptions of Theorem 1.1, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M257">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M258">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M259">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M260">View MathML</a>and the constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M261">View MathML</a>does not depend onu, wr.

Proof This follows from (H5) if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M262">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M263">View MathML</a>. □

Lemma 2.9The sequences given in Lemma 2.7 are bounded.

Proof Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M264">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M265">View MathML</a>. Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M266">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M267">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M268">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M269">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M270">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M271">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M272">View MathML</a> in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M273">View MathML</a> a.e. on ℝ, after passing to a subsequence.

Case 1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M274">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M275">View MathML</a> be the subset of ℝ where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M276">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M277">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M278">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M275">View MathML</a>. It follows from (H4) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M281">View MathML</a>

which together with Lemmas 2.3 and 2.7 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M282">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M283">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M284">View MathML</a> (by (2.1)) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M285">View MathML</a>

It is a contradiction.

Case 2. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M286">View MathML</a>. We claim that there is a constant C independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M287">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M288">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M289">View MathML</a>

(2.9)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M290">View MathML</a>

it follows from the definition of I that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M291">View MathML</a>

(2.10)

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M292">View MathML</a> in (2.10), then it follows from Lemma 2.8 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M293">View MathML</a>

Thus (2.9) holds.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M294">View MathML</a> be a fixed constant and take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M295">View MathML</a>

Therefore, (2.9) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M296">View MathML</a>

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M268">View MathML</a> and Lemma 2.7 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M298">View MathML</a>

(2.11)

Note that Lemmas 2.3 and 2.7 and (H4) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M299">View MathML</a>

It follows from the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M300">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M301">View MathML</a> due to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M53">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M303">View MathML</a>

(2.12)

for all sufficiently large n. We take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M304">View MathML</a>, by (2.12) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M270">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M306">View MathML</a>

(2.13)

for all sufficiently large n. By (H1) and (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M307">View MathML</a>

(2.14)

For all sufficiently large n, by (2.13) and (2.14), it follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M309">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M283">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M284">View MathML</a> (by (2.1)) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M312">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M313">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M314">View MathML</a>, contrary to (2.11).

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167">View MathML</a> are bounded. The proof is finished. □

3 Proofs of the main results

Proof of Theorem 1.1 From Lemma 2.7, there are sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M316">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M253">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M318">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M319">View MathML</a>. By Lemma 2.9, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167">View MathML</a> is bounded in E. Thus we can assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M321">View MathML</a> in E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M322">View MathML</a> a.e. on ℝ. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M323">View MathML</a>

Hence, in the limit,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M324">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M325">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M326">View MathML</a>

(3.1)

Similar to (2.7) and (2.8), we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M327">View MathML</a>

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M325">View MathML</a>, (3.1) and Lemma 2.3 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M329">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M330">View MathML</a>. □

Proof of Theorem 1.2 By Theorem 1.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M331">View MathML</a>, where ℳ is the collection of solutions of (1.1). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M332">View MathML</a>

If u is a solution of (1.1), then by Lemma 2.8 (take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M333">View MathML</a>),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M334">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M335">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167">View MathML</a> be a sequence in ℳ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M337">View MathML</a>

(3.2)

By Lemma 2.9, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167">View MathML</a> is bounded in E. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M321">View MathML</a> in E, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M179">View MathML</a> a.e. on ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M322">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M69">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M343">View MathML</a> (by (2.1)), after passing to a subsequence. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M344">View MathML</a>

Hence, in the limit,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M345">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M325">View MathML</a>. Similar to (2.7) and (2.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M347">View MathML</a>

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M325">View MathML</a> and (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M349">View MathML</a>

Now suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M350">View MathML</a>

It follows from (H1) that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M351">View MathML</a>, there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M352">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M353">View MathML</a>

(3.3)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M354">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M355">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M167">View MathML</a> be a sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M60">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M358">View MathML</a>

(3.4)

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M359">View MathML</a>

which together with (3.3), Hölder’s inequality and the Sobolev embedding theorem implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M360">View MathML</a>

(3.5)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M361">View MathML</a>

(3.6)

From (3.5) and (3.6), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M362">View MathML</a>

which means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M363">View MathML</a> for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M364">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M322">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M69">View MathML</a>, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M330">View MathML</a>. As before, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M368">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/106/mathml/M301">View MathML</a>. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The main idea of this paper was proposed by G-WC and G-WC prepared the manuscript initially and JW performed a part of steps of the proofs in this research. All authors read and approved the final manuscript.

Acknowledgements

The authors thank the referees and the editors for their helpful comments and suggestions. Research was supported by the Tianyuan Fund for Mathematics of NSFC (Grant No. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant No. 13A110015).

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