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Solutions of perturbed p-Laplacian equation with critical nonlinearity and magnetic fields

Huixing Zhang* and Juan Jiang

Author Affiliations

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

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Boundary Value Problems 2013, 2013:217  doi:10.1186/1687-2770-2013-217


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/217


Received:3 April 2013
Accepted:28 August 2013
Published:5 November 2013

© 2013 Zhang and Jiang; licensee Springer.

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

Abstract

In this paper, we consider a perturbed p-Laplacian equation with critical nonlinearity and magnetic fields on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M1">View MathML</a>. By using the variational method, we establish the existence of nontrivial solutions of the least energy.

MSC: 35B33, 35J60, 35J65.

Keywords:
p-Laplacian equation; critical nonlinearity; magnetic fields; mountain pass theorem

1 Introduction

In this paper, we are concerned with the existence of nontrivial solutions to the following perturbed p-Laplacian equation with critical nonlinearity and magnetic fields of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M2">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M3">View MathML</a>, i is the imaginary unit, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M4">View MathML</a> is a real vector potential, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M6">View MathML</a> denotes the Sobolev critical exponent and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M7">View MathML</a>.

This paper is motivated by some works concerning the nonlinear Schr?dinger equation with magnetic fields of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M8">View MathML</a>

(1.2)

where ? is Planck?s constant, i is the imaginary unit, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M9">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M10">View MathML</a>) is the critical exponent, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M11">View MathML</a> is a real vector potential, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M13">View MathML</a> is a scalar electric potential.

In physics, we are interested in the standing wave solutions, that is, solutions to (1.2) of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M14">View MathML</a>

where ? is a sufficiently small constant, E is a real number, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M15">View MathML</a> is a complex-valued function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M16">View MathML</a>

(1.3)

We can conduct the transition from quantum mechanics to classical mechanics by letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M17">View MathML</a>. Thus, the existence of semiclassical solutions has a great charm in physical interest.

Problem (1.3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M18">View MathML</a> has an extensive literature. Different approaches have been taken to investigate this problem under various hypotheses on the potential and nonlinearity. See for example [1-18] and the references therein. The above-mentioned papers mostly concentrated on the nonlinearities with subcritical conditions. Floer and Weinstein in [11] first studied the existence of single and multiple spike solutions based on the Lyapunov-Schmidt reductions. Subsequently, Oh [16,17] extended the results in a higher dimension. Kang and Wei [14] established the existence of positive solutions with any prescribed number of spikes, clustering around a given local maximum point of the potential function. In accordance with the Sobolev critical nonlinearities, there have been many papers devoted to studying the existence of solutions to elliptic boundary-valued problems on bounded domains after the pioneering work by Br?zis and Nirenberg [4]. Ding and Lin [8] first studied the existence of semi-classical solutions to the problem on the whole space with critical nonlinearities and established the existence of positive solutions, as well as of those that change sign exactly once. They also obtained multiplicity of solutions when the nonlinearity is odd.

As far as problem (1.3) in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M19">View MathML</a> is concerned, we recall Bartsch [2], Cingolani [5] and Esteban and Lions [10]. This kind of paper first appeared in [10]. The authors obtained the existence results of standing wave solutions for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M20">View MathML</a> and special classes of magnetic fields. Cingolani [5] proved that the magnetic potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M21">View MathML</a> only contributes to the phase factor of the solitary solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M20">View MathML</a> sufficiently small. For more results, we refer the reader to [19-21] and the references therein.

For general <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M23">View MathML</a>, most of the works studied the existence results to equation (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M18">View MathML</a>. See, for example, [22-28] and the references therein. These papers are mostly devoted to the study of the existence of solutions to the problem on bounded domains with the Sobolev subcritical nonlinearities.

However, to our best knowledge, it seems that there is no work on the existence of semiclassical solutions to perturbed p-Laplacian equation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M1">View MathML</a> involving critical nonlinearity and magnetic fields. In this paper, we consider problem (1.1) with magnetic fields. The main difficulty in the case is the lack of compactness of the energy functional associated to equation (1.1) because of unbounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M1">View MathML</a> and critical nonlinearity. At the same time, we must consider complex-valued functions for the appearance of electromagnetic potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M21">View MathML</a>. To overcome this difficulty, we chiefly follow the ideas of [5]. Notice that although the ideas were used in other problems, the adaption of the procedure to our problem is not trivial at all. We need to make careful and complex estimates and prove that the energy functional possesses a Palais-Smale sequence, which has a strongly convergent sequence.

We make the following assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M31">View MathML</a> throughout the paper:

(V0) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M33">View MathML</a>, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M34">View MathML</a> such that the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M35">View MathML</a> has a finite Lebesgue measure;

(A0) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M37">View MathML</a>;

(K0) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M39">View MathML</a>;

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M41">View MathML</a> uniformly in x as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M42">View MathML</a>;

(H2) there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M44">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M45">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M46">View MathML</a>;

(H3) there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M49">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M51">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M46">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M53">View MathML</a>.

Our main result is the following.

Theorem 1Assume that (V0), (A0), (K0) and (H1)-(H3) hold. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M54">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M55">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M56">View MathML</a>, equation (1.1) has at least one positive least energy solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M57">View MathML</a>, which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M58">View MathML</a>

The paper is organized as follows. In Section?2, we give some necessary preliminaries. Section?3 is devoted to the technical lemmas. The proof of Theorem?2 is given in the last section.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M59">View MathML</a>. Equation (1.1) reads then as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M60">View MathML</a>

(2.1)

We are going to prove the following result.

Theorem 2Assume that (V0), (A0), (K0) and (H1)-(H3) are satisfied. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M54">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M62">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M63">View MathML</a>, then equation (2.1) has at least one solution of least energy<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M64">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M65">View MathML</a>

(2.2)

In order to prove these theorems, we introduce the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M66">View MathML</a>

equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M67">View MathML</a>

It is known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a> is the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M69">View MathML</a>. Similar to the diamagnetic inequality [10], we have the following inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M70">View MathML</a>

In fact, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M21">View MathML</a> is real-valued, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M72">View MathML</a>

(2.3)

(the bar denotes a complex conjugation). This inequality implies that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M73">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M74">View MathML</a>, and, therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M75">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M76">View MathML</a>. That is, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M77">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M80">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M81">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M1">View MathML</a>.

Solutions of (2.1) will be sought in the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a> as critical points of the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M85">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M86">View MathML</a>.

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M88">View MathML</a>-functional on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a>[29].

3 Behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90">View MathML</a> sequence and a mountain pass structure

In this section, we commence by establishing the necessary results which complete the proof of Theorem?2.

Lemma 3.1Let (V0), (A0), (K0) and (H1)-(H3) be satisfied. For the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90">View MathML</a>sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M92">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87">View MathML</a>, we get that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M94">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M95">View MathML</a>is bounded in the space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a>.

Proof Under assumptions (K0) and (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M97">View MathML</a>

In connection with the facts that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M99">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M100">View MathML</a>, we obtain that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90">View MathML</a> sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M95">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a>, and the energy level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M94">View MathML</a>.??

Next, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M95">View MathML</a> denote a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90">View MathML</a> sequence. By Lemma?3.1, it is bounded, thus, without loss of generality, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M77">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a>. Furthermore, passing to a subsequence, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M80">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M1">View MathML</a>.

Lemma 3.2For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M114">View MathML</a>, there is a subsequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M115">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M116">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M117">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M118">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M119">View MathML</a>.

Proof It is easily obtained by the similar proof of Lemma?3.2 [8].??

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M120">View MathML</a> be a smooth function satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M122">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M123">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M124">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M125">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M126">View MathML</a>. It is not difficult to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M127">View MathML</a>

Lemma 3.3One has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M128">View MathML</a>

uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M129">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M130">View MathML</a>.

Proof By direct computation, we easily obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M131">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a>. The local compactness of the Sobolev embedding implies that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M133">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M134">View MathML</a>

uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M130">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M116">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M137">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M138">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M139">View MathML</a>. By the assumptions and the H?lder inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M140">View MathML</a>

This proof is completed.??

Lemma 3.4One has along a subsequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M141">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M142">View MathML</a>

Proof Combining Lemma?2.1 of [30] and the arguments of [31], one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M143">View MathML</a>

By the Br?zis-Lieb lemma [32], we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M144">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M145">View MathML</a>

We now observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M147">View MathML</a>, which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M148">View MathML</a>

Moreover, by direct computation, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M149">View MathML</a>

It then follows from the standard arguments that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M150">View MathML</a>

uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M130">View MathML</a>. Combining Lemma?3.3, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M152">View MathML</a>. The proof is completed.??

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M153">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M154">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M157">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a>.

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M159">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M160">View MathML</a>. Together with Lemma?3.4, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M161">View MathML</a>

(3.1)

In the following, we consider the energy level of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87">View MathML</a> below which the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90">View MathML</a> condition holds.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M164">View MathML</a>, where b is the positive constant in assumption (V0). Since the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M165">View MathML</a> has a finite measure, combining the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M157">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M167">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M168">View MathML</a>

(3.2)

Furthermore, by (K0) and (H1)-(H3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M169">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M170">View MathML</a>

(3.3)

Let S be the best Sobolev constant of the immersion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M171">View MathML</a>

Lemma 3.5There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M172">View MathML</a> (independent of?) such that, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90">View MathML</a>sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M92">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M77">View MathML</a>, either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M79">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M179">View MathML</a>.

Proof Arguing by contradiction, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M180">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M181">View MathML</a>

Combining the Sobolev inequality, (3.2) and (3.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M182">View MathML</a>

which further gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M183">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M184">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M185">View MathML</a>

We obtain the desired conclusion.??

Lemma 3.6There exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M172">View MathML</a> (independent of?) such that if a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M90">View MathML</a>sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M92">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M190">View MathML</a>, the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M95">View MathML</a>has a strongly convergent subsequence in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M68">View MathML</a>.

Proof By the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M193">View MathML</a> and Lemma?3.5, we easily get the required conclusion.??

Now, we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M194">View MathML</a> and prove that the energy functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87">View MathML</a> possesses the mountain pass structure.

Lemma 3.7Under the assumptions of Theorem?2, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M196">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M197">View MathML</a>

Proof The proof of Lemma?3.7 is similar to the one of Lemma?4.1 in [8].??

Lemma 3.8For any finite dimensional subspace<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M198">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M199">View MathML</a>

Proof By assumptions (K0) and (H3), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M200">View MathML</a>

Since all norms in a finite-dimensional space are equivalent, in connection with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M48">View MathML</a>, we obtain the desired conclusion.??

For ? large enough and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M202">View MathML</a> small sufficiently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M204">View MathML</a> condition by Lemma?3.6. Furthermore, we will find special finite-dimensional subspace, by which we establish sufficiently small minimax levels.

Define the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M205">View MathML</a>

It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M206">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M207">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M73">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M209">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M210">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M211">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M213">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M214">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M215">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M216">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M217">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M218">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M219">View MathML</a>

We derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M220">View MathML</a>

Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M222">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M214">View MathML</a>. Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M224">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M225">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M226">View MathML</a>

(3.4)

Lemma 3.9Under the assumptions of Theorem?2, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M54">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M228">View MathML</a>such that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M229">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M230">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M232">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M233">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M234">View MathML</a>is defined in Lemma?3.7.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M54">View MathML</a>, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M236">View MathML</a> so small that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M237">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M215">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M239">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M240">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M241">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M242">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M243">View MathML</a>. Then, combining (3.4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M244">View MathML</a> meets the requirements.??

4 Proof of Theorem 2

In this section, we give the proof of Theorem?2.

Proof By Lemma?3.9, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M54">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M246">View MathML</a>, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M228">View MathML</a> and define the minimax value

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M248">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M249">View MathML</a>.

Lemma?3.6 shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M87">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M204">View MathML</a> condition. Therefore, by the mountain pass theorem, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M252">View MathML</a>, which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M253">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M254">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M64">View MathML</a> is a weak solution of (2.1). Furthermore, it is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M64">View MathML</a> is the least energy solution of equation (2.1).

Moreover, together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M257">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M254">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M259">View MathML</a>

By inequality (2.3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/217/mathml/M260">View MathML</a>

The proof is complete.??

Competing interests

The authors declare that they have no competing interests.

Authors? contributions

The authors contributed equally in this article. They 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 was supported by the National Natural Science Foundation of China (11271364) and the Fundamental Research Funds for the Central Universities (2012QNA46).

References

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