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Existence of nontrivial solutions for perturbed p-Laplacian system involving critical nonlinearity and magnetic fields

Huixing Zhang*, Jiaying Liu and Wenbin Liu

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Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, People’s Republic of China

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Citation and License

Boundary Value Problems 2013, 2013:11  doi:10.1186/1687-2770-2013-11


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/11


Received:5 October 2012
Accepted:9 January 2013
Published:24 January 2013

© 2013 Zhang et al.; licensee Springer

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

Abstract

Under the suitable assumptions, we establish the existence of nontrivial solutions for a perturbed p-Laplacian system in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M1">View MathML</a> with critical nonlinearity and magnetic fields by using the variational method.

MSC: 35B33, 35J60, 35J65.

Keywords:
p-Laplacian system; critical nonlinearity; magnetic fields; variational method

1 Introduction

In this paper, we consider a class of quasi-linear elliptic systems of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M2">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M3">View MathML</a>, i is the imaginary unit, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M4">View MathML</a> is real vector potential, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M6">View MathML</a> is a non-negative potential, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M7">View MathML</a> denotes the Sobolev critical exponent for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M9">View MathML</a> is a bounded positive coefficient.

The scalar case corresponding to (1.1) has received considerable attention in recent years. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M11">View MathML</a>, the scalar case corresponding to (1.1) turns into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M12">View MathML</a>

(1.2)

The equation (1.2) arises in finding standing wave solutions of the nonlinear Schrödinger equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M13">View MathML</a>

(1.3)

A standing wave solution of (1.3) is a solution of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M14">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M15">View MathML</a> solves (1.3) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M16">View MathML</a> solves (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M18">View MathML</a>.

The equation (1.2) has been extensively investigated in the literature based on various assumptions of the potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M6">View MathML</a> and the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M20">View MathML</a>. See, for example, [1-15] and the references therein.

There are also many works dealing with the magnetic fields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M10">View MathML</a> for the scalar case corresponding to (1.1). In [16], the authors firstly obtained the existence of standing waves for special classes of magnetic fields. For many results, we refer the reader to [17-22].

For general <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M23">View MathML</a>, most of the work, as we know, consider the scalar case which corresponds to (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M11">View MathML</a>. See [23-27] and the references therein. We especially mention [24] for the existence of positive solutions for a class of p-Laplacian equations. Gloss [24] studied the existence and asymptotic behavior of positive solutions for quasi-linear elliptic equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M25">View MathML</a>

(1.4)

where f is a subcritical nonlinearity without some growth conditions such as the Ambrosetti-Rabinowitz condition. The problem (1.4) has also been studied in [28-32]. The main difficulty in treating this class of equation (1.4) is a possible lack of compactness due to the unboundedness of the domain.

However, to our best knowledge, it seems there is almost no work on the existence of non-trivial solutions to the problem (1.1) involving critical nonlinearity and magnetic fields. We mainly follow the idea of [7]. Observe that though the idea was used in other problems, the adaption of the procedure to the problem is not trivial at all. Because of the appearance of magnetic fields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M26">View MathML</a>, we must deal with the problem for complex-valued functions and therefore we need more delicate estimates.

The outline of the paper is as follows. The forthcoming section is the main result and preliminary results including the appropriate space setting to work with. In Section 3, we study the behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27">View MathML</a> sequence. Section 4 gets that the functional associated to the problem possesses the mountain geometry structure, and the last section concludes the proof of the main result.

2 Main results and preliminaries

Firstly, we make the following assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M9">View MathML</a> throughout the paper:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M32">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M34">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M35">View MathML</a> such that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M36">View MathML</a> has finite Lebesgue measure;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M37">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M39">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M40">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M42">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M43">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M45">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M46">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M47">View MathML</a>) there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M49">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M50">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M51">View MathML</a>) there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M54">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M56">View MathML</a>.

Under the above mentioned conditions, we get the following result.

Theorem 1Suppose that the assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M32">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M37">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M40">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M43">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M51">View MathML</a>) hold. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M62">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M63">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M64">View MathML</a>, the problem (1.1) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M65">View MathML</a>which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M66">View MathML</a>

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M67">View MathML</a>, the problem (1.1) is equivalent to the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M68">View MathML</a>

(2.1)

We are going to prove the following result.

Theorem 2Suppose that the assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M32">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M37">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M40">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M43">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M51">View MathML</a>) hold. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M62">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M75">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M76">View MathML</a>, the problem (2.1) has at least one solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M77">View MathML</a>which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M78">View MathML</a>

(2.2)

For convenience, we quote the following notations. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M79">View MathML</a> denote the Banach space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M80">View MathML</a>

equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M81">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M83">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M84">View MathML</a>.

Similar to the diamagnetic inequality [16], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M85">View MathML</a>

(2.3)

(the bar denotes complex conjugation). This inequality shows that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M86">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M87">View MathML</a> and therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M88">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M89">View MathML</a>. That is to say, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M90">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M79">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M92">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M93">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M92">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M96">View MathML</a>.

The energy functional associated with (2.1) is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M97">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M98">View MathML</a>.

Under the assumptions of Theorem 2, standard arguments [33] show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M99">View MathML</a> and its critical points are weak solutions of the equation (2.1).

3 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27">View MathML</a> condition

We call a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M101">View MathML</a> a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27">View MathML</a> sequence if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M104">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M105">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M105">View MathML</a> is the dual space of E). <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107">View MathML</a> is said to satisfy the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27">View MathML</a> condition if any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27">View MathML</a> sequence contains a convergent subsequence.

The main result of Section 3 is the following compactness result.

Proposition 3.1Let the assumptions of Theorem 2 be satisfied. There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M110">View MathML</a>independent ofλsuch that, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27">View MathML</a>sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M101">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M114">View MathML</a>, either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M115">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M116">View MathML</a>.

As a consequence, we obtain the following result.

Proposition 3.2Assume that the assumptions of Proposition 3.1 hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M117">View MathML</a>satisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27">View MathML</a>condition for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M119">View MathML</a>.

In order to prove Proposition 3.1, we need the following lemmas.

Lemma 3.1Let the assumptions of Theorem 2 be satisfied. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M120">View MathML</a>is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27">View MathML</a>sequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M123">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M120">View MathML</a>is bounded in the spaceE.

Proof

One has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M125">View MathML</a>

Together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M127">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M128">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M129">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M120">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M123">View MathML</a>. □

From Lemma 3.1, we may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M114">View MathML</a> in E and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M115">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M134">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M137">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M96">View MathML</a>.

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M139">View MathML</a>. There is a subsequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M140">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M141">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M142">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M143">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M144">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M145">View MathML</a>.

Proof The proof of Lemma 3.2 is similar to that of Lemma 3.2 of [27], so we omit it. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M146">View MathML</a> be a smooth function satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M147">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M148">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M149">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M150">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M151">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M153">View MathML</a>. Obviously, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M154">View MathML</a>

(3.1)

Lemma 3.3One has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M155">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M156">View MathML</a>

uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M157">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M158">View MathML</a>.

Proof The local compactness of Sobolev embedding implies that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M159">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M160">View MathML</a>

uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M161">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M162">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M142">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M164">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M143">View MathML</a>. Together with the assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M47">View MathML</a>) and the Hölder inequality, it follows from Lemma 3.2 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M167">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M168">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M169">View MathML</a>) are positive constants. Similarly, we can prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M170">View MathML</a>

 □

Lemma 3.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M120">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M172">View MathML</a>be as defined above. Then the following conclusions hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M173">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M174">View MathML</a>

Proof By using the similar arguments of [34,35], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M175">View MathML</a>

By (3.1) and the similar idea of proving the Brézis-Lieb lemma [36], it is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M176">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M177">View MathML</a>

Furthermore, using the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M179">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M180">View MathML</a>

In order to prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M181">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M182">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M157">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M184">View MathML</a>

It is standard to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M185">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M186">View MathML</a>

uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M157">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M158">View MathML</a>. Together with Lemma 3.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M189">View MathML</a>

 □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M190">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M191">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M192">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M193">View MathML</a>. From (3.1), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M115">View MathML</a> in E if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M195">View MathML</a> in E.

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M196">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M197">View MathML</a>. Furthermore, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M198">View MathML</a>

(3.2)

Now, we consider the energy level of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107">View MathML</a> below which the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M27">View MathML</a> condition holds.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M201">View MathML</a>, where b is a positive constant in the assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M32">View MathML</a>). Since the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M203">View MathML</a> has finite measure, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M204">View MathML</a>

(3.3)

In connection with the assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M43">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M51">View MathML</a>) and the Young inequality, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M207">View MathML</a> such that

(3.4)

Let S be the best Sobolev constant of the immersion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M209">View MathML</a>

Proof of Proposition 3.1 Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M210">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M211">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M212">View MathML</a>

By the Sobolev embedding inequality and the diamagnetic inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M213">View MathML</a>

This, together with (3.2), gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M214">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M215">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M216">View MathML</a>

This completes the proof of Proposition 3.1. □

Proof of Proposition 3.2 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M119">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M218">View MathML</a>

In connection with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M219">View MathML</a> and Proposition 3.1, we complete this proof. □

4 The mountain-pass structure

In the following, we always consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M220">View MathML</a>. We will prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107">View MathML</a> possesses the mountain-pass structure which has been carefully discussed in the works [37,38].

Lemma 4.1Let the assumptions of Theorem 2 be satisfied. There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M222">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M223">View MathML</a>

Proof By (3.4), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M224">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M225">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M226">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M227">View MathML</a>

In connection with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M228">View MathML</a>, we may choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M229">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M230">View MathML</a>

The fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M231">View MathML</a> implies the desired conclusion. □

Lemma 4.2Under the assumptions of Lemma 4.1, for any finite dimensional subspace<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M232">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M233">View MathML</a>

Proof Together with the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M234">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M235">View MathML</a>

Since all norms in a finite-dimensional space are equivalent and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M236">View MathML</a>, we complete the proof. □

In the following, we will find special finite-dimensional subspaces by which we establish sufficiently small mini-max levels.

Define the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M237">View MathML</a>

Obviously, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M238">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M239">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M84">View MathML</a>.

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M241">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M242">View MathML</a>

Then, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M224">View MathML</a>, there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M244">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M245">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M246">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M247">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M248">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M249">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M250">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M251">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M252">View MathML</a>

By direct computation, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M253">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M255">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M256">View MathML</a>, we know that there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M257">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M258">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M259">View MathML</a>

(4.1)

Lemma 4.3For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M260">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M75">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M262">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M263">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M265">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M266">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M267">View MathML</a>is defined in Lemma 4.1.

Proof This proof is similar to that of Lemma 4.3 in [7], so we omit the details. □

5 Proof of Theorem 2

Proof By using Lemma 4.3, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M260">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M269">View MathML</a>, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M75">View MathML</a> and define the mini-max level

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M271">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M272">View MathML</a>.

By Proposition 3.1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M107">View MathML</a> satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M274">View MathML</a> condition. Hence, by the mountain-pass theorem, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M275">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M276">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M277">View MathML</a>. This shows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M278">View MathML</a> is a weak solution of (2.1).

Moreover, note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M279">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M277">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M281">View MathML</a>

Furthermore, together with the diamagnetic inequality, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M278">View MathML</a> satisfies the estimate (2.2). The proof is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally in this article. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to appreciate the referees for their precious comments and suggestions about the original manuscript. This research is supported by the National Natural Science Foundation of China (11271364) and the Fundamental Research Funds for the Central Universities (2012QNA46).

References

  1. Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text OpenURL

  2. Benci, V: On critical point theory of indefinite functions in the presence of symmetries. Trans. Am. Math. Soc.. 274, 533–572 (1982). Publisher Full Text OpenURL

  3. Brézis, H, Nirenberg, L: Positive solutions of nonlinear elliptic equation involving critical Sobolev exponents. Commun. Pure Appl. Math.. 16, 437–477 (1983)

  4. Cingolani, S, Nolasco, M: Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equation. Proc. R. Soc. Edinb.. 128, 1249–1260 (1998). Publisher Full Text OpenURL

  5. Del Pino, M, Felmer, PL: Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal.. 149, 245–265 (1997). Publisher Full Text OpenURL

  6. Del Pino, M, Felmer, PL: Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 15, 127–149 (1998). Publisher Full Text OpenURL

  7. Ding, YH, Lin, FH: Solutions of perturbed Schrödinger equations with critical nonlinearity. Calc. Var.. 30, 231–249 (2007). Publisher Full Text OpenURL

  8. Guedda, M, Veron, L: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal.. 12, 879–902 (1989)

  9. Jeanjean, L, Tanaka, K: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ.. 21, 287–318 (2004)

  10. Kang, X, Wei, JC: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differ. Equ.. 5, 899–928 (2000)

  11. Li, YY: On a singularly perturbed elliptic equation. Adv. Differ. Equ.. 2, 955–980 (1997)

  12. Oh, YG: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys.. 131, 223–253 (1990). Publisher Full Text OpenURL

  13. Oh, YG: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M283">View MathML</a>. Commun. Partial Differ. Equ.. 13, 1499–1519 (1988). Publisher Full Text OpenURL

  14. Pistoia, A: Multi-peak solutions for a class of some results on a class of nonlinear Schrödinger equations. Nonlinear Differ. Equ. Appl.. 9, 69–91 (2002). Publisher Full Text OpenURL

  15. Floer, A, Weinstein, A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal.. 69, 397–408 (1986). PubMed Abstract | Publisher Full Text OpenURL

  16. Esteban, M, Lions, PL: Stationary solutions of nonlinear Schrödinger equation with an external magnetic field. Partial Differential Equations and the Calculus of Variations, Essays in Honor of E. De Giorgi, pp. 369–408. Brikhäuser, Basel (1990)

  17. Arioli, G, Szulkin, A: A semilinear Schrödinger equation in the presence of a magnetic field. Arch. Ration. Mech. Anal.. 170, 277–295 (2003). Publisher Full Text OpenURL

  18. Bartsch, T, Dancer, EN, Peng, S: On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields. Adv. Differ. Equ.. 11, 781–812 (2006)

  19. Cao, D, Tang, Z: Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields. J. Differ. Equ.. 222, 381–424 (2006). Publisher Full Text OpenURL

  20. Cingolani, S: Semiclassical stationary states of nonlinear Schrödinger equation with an external magnetic field. J. Differ. Equ.. 188, 52–79 (2003). PubMed Abstract | Publisher Full Text OpenURL

  21. Han, P: Solutions for singular critical growth Schrödinger equation with magnetic field. Port. Math.. 63, 37–45 (2006)

  22. Kurata, K: Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields. Nonlinear Anal.. 41, 763–778 (2000). Publisher Full Text OpenURL

  23. Alves, CO, Ding, YH: Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity. J. Math. Anal. Appl.. 279, 508–521 (2003). Publisher Full Text OpenURL

  24. Gloss, E: Existence and concentration of bound states for a p-Laplacian equation in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M96">View MathML</a>. Adv. Nonlinear Stud.. 10, 273–296 (2010)

  25. Liu, CG, Zheng, YQ: Existence of nontrivial solutions for p-Laplacian equations in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M96">View MathML</a>. J. Math. Anal. Appl.. 380, 669–679 (2011). Publisher Full Text OpenURL

  26. Manásevich, R, Mawhin, J: Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators. J. Korean Math. Soc.. 5, 665–685 (2000)

  27. Zhang, HX, Liu, WB: Existence of nontrivial solutions to perturbed p-Laplacian system in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/11/mathml/M96">View MathML</a> involving critical nonlinearity. Bound. Value Probl.. 2012, Article ID 53 (2012)

  28. Dinu, TL: Entire solutions of Schrödinger systems with discontinuous nonlinearity and sign-changing potential. Math. Model. Anal.. 13(3), 229–242 (2006)

  29. Dinu, TL: Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev space with variable exponent. Nonlinear Anal.. 65(7), 1414–1424 (2006). Publisher Full Text OpenURL

  30. Gazzola, F, Radulescu, V: A nonsmooth critical point theory approach to some nonlinear elliptic equations in unbounded domains. Differ. Integral Equ.. 13, 47–60 (2000)

  31. Ghanmi, A, Maagli, H, Radulescu, V, Zeddini, N: Large and bounded solutions for a class of nonlinear Schrödinger stationary systems. Anal. Appl.. 7, 391–404 (2009). Publisher Full Text OpenURL

  32. Rabinowitz, PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys.. 43, 270–291 (1992). Publisher Full Text OpenURL

  33. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems, Springer, New York (1989)

  34. Ghoussoub, N, Yuan, C: Multiple solutions for quasi-linear PDES involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc.. 352, 5703–5743 (2000). Publisher Full Text OpenURL

  35. Li, YY, Guo, QQ, Niu, PC: Global compactness results for quasilinear elliptic problems with combined critical Sobolev-Hardy terms. Nonlinear Anal.. 74, 1445–1464 (2011). Publisher Full Text OpenURL

  36. Brézis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functional. Proc. Am. Math. Soc.. 88, 486–490 (1983)

  37. Pucci, P, Radulescu, V: The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital., Ser. IX. 3, 543–584 (2010)

  38. Radulescu, V: Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic and Variational Methods, Hindawi Publ. Corp, New York (2008)