Open Access Research

Multiple positive solutions for semilinear elliptic systems involving subcritical nonlinearities in R N

Huei-li Lin

Author Affiliations

Department of Natural Sciences in the Center for General Education, Chang Gung University, Tao-Yuan, 333, Taiwan, R.O.C

Boundary Value Problems 2012, 2012:118  doi:10.1186/1687-2770-2012-118


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/118


Received:29 March 2012
Accepted:4 October 2012
Published:24 October 2012

© 2012 Lin; licensee Springer

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

Abstract

In this paper, we investigate the effect of the coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M2">View MathML</a> of the subcritical nonlinearity. Under some assumptions, for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M3">View MathML</a>, there are at least k (≥1) positive solutions of the semilinear elliptic systems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M4">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M7">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M8">View MathML</a>.

MSC: 35J20, 35J25, 35J65.

Keywords:
semilinear elliptic systems; subcritical exponents; Nehari manifold

1 Introduction

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M12">View MathML</a>, we consider the semilinear elliptic systems

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M3">View MathML</a>.

Let f, g and h satisfy the following conditions:

(A1) f is a positive continuous function in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M16">View MathML</a>.

(A2) there exist k points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M17">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M1">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M19">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M20">View MathML</a>.

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M21">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M22">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M23">View MathML</a>.

In [1], if Ω is a smooth and bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M24">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M25">View MathML</a>), they considered the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M26">View MathML</a>

and proved the existence of a least energy solution in Ω for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M28">View MathML</a>. Lin and Wei also showed that this system has a least energy solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M1">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M31">View MathML</a>. In this paper, we study the effect of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M32">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M33">View MathML</a>). Recently, many authors [2-5] considered the elliptic systems with subcritical or critical exponents, and they proved the existence of a least energy positive solution or the existence of at least two positive solutions for these problems. In this paper, we construct the k compact Palais-Smale sequences which are suitably localized in correspondence of k maximum points of f. Then we could show that under some assumptions (A1)-(A3), for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M3">View MathML</a>, there are at least k (≥1) positive solutions of the elliptic system (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>). By the change of variables

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M36">View MathML</a>

System (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M33">View MathML</a>) is transformed to

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M39">View MathML</a> be the space with the standard norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M40">View MathML</a>

Associated with the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>), we consider the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M42">View MathML</a>-functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M44">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M45">View MathML</a>

Actually, the weak solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M46">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>) is the critical point of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M48">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M46">View MathML</a> satisfies

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M51">View MathML</a>.

We consider the Nehari manifold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M52">View MathML</a>

(1.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M53">View MathML</a>

The Nehari manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54">View MathML</a> contains all nontrivial weak solutions of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>).

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M56">View MathML</a>

(1.2)

then by [[2], Theorem 5], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M57">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M59">View MathML</a> is the best Sobolev constant defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M60">View MathML</a>

For the semilinear elliptic systems (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M61">View MathML</a>)

we define the energy functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M63">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M64">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M65">View MathML</a> (=1), then we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M66">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M67">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M68">View MathML</a>.

It is well known that this problem

has the unique, radially symmetric and positive ground state solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M70">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M72">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M73">View MathML</a>

Moreover, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M74">View MathML</a>

This paper is organized as follows. First of all, we study the argument of the Nehari manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54">View MathML</a>. Next, we prove that the existence of a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M76">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>). Finally, in Section 4, we show that the condition (A2) affects the number of positive solutions of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>); that is, there are at least k critical points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M79">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M81">View MathML</a> ((PS)-value) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82">View MathML</a>.

Theorem 1.1 (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>) has at least one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M84">View MathML</a>, that is, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M33">View MathML</a>) admits at least one positive solution.

Theorem 1.2There exist two positive numbers<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M86">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M87">View MathML</a>such that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>) has at leastkpositive solutions for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M89">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M90">View MathML</a>, that is, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M33">View MathML</a>) admits at leastkpositive solutions.

2 Preliminaries

By studying the argument of Han [[7], Lemma 2.1], we obtain the following lemma.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M92">View MathML</a> (possibly unbounded) be a smooth domain. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M94">View MathML</a>weakly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M95">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M97">View MathML</a>almost everywhere in Ω, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M98">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a> is not bounded from below in H. From the following lemma, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a> is bounded from below on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54">View MathML</a>.

Lemma 2.2The energy functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a>is bounded from below on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M104">View MathML</a>, by (1.1), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M105">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M58">View MathML</a>. Hence, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M107">View MathML</a> is bounded from below on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54">View MathML</a>. □

We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M109">View MathML</a>

Lemma 2.3 (i) There exist positive numbersσand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M110">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M111">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M112">View MathML</a>;

(ii) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M113">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M114">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M115">View MathML</a>.

Proof (i) By (1.2), the Hölder inequality (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M117">View MathML</a>) and the Sobolev embedding theorem, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M118">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M120">View MathML</a>. Hence, there exist positive σ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M110">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M122">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M123">View MathML</a>.

(ii) For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M124">View MathML</a>, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M125">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M126">View MathML</a>. Fix some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M124">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M128">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M130">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M131">View MathML</a>. □

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M132">View MathML</a>

Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M104">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M134">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M135">View MathML</a>

(2.2)

Lemma 2.4For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M136">View MathML</a>, there exists a unique positive number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M137">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M138">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M139">View MathML</a>.

Proof Fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M136">View MathML</a>, we consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M141">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M142">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M143">View MathML</a>, by Lemma 2.3(i), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M144">View MathML</a> is achieved at some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M145">View MathML</a>. Moreover, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M146">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M138">View MathML</a>. Next, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M137">View MathML</a> is a unique positive number such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M146">View MathML</a>. Consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M150">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M151">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M152">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M153">View MathML</a>

there exists a unique positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M154">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M155">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M156">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M157">View MathML</a>. □

Remark 2.5 By Lemma 2.3(i) and Lemma 2.4, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M158">View MathML</a> for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M110">View MathML</a>.

Lemma 2.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M160">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M161">View MathML</a>

then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M84">View MathML</a>is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>).

Proof By (2.2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M164">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M165">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M166">View MathML</a>, by the Lagrange multiplier theorem, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M167">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M168">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M170">View MathML</a>

It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M172">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M84">View MathML</a> is a nontrivial solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M176">View MathML</a>. □

3 (PS)γ-condition in H for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M177">View MathML</a>

First of all, we define the Palais-Smale (denoted by (PS)) sequence and (PS)-condition in H for some functional J.

Definition 3.1 (i) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M178">View MathML</a>, a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M179">View MathML</a> is a (PS)γ-sequence in H for J if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M180">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M181">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169">View MathML</a> is the dual space of H;

(ii) J satisfies the (PS)γ-condition in H if every (PS)γ-sequence in H for J contains a convergent subsequence.

Applying Ekeland’s variational principle and using the same argument as in Cao-Zhou [8] or Tarantello [9], we have the following lemma.

Lemma 3.2 (i) There exists a (PS)-sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M186">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M188">View MathML</a>.

In order to prove the existence of positive solutions, we want to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a> satisfies the (PS)γ-condition in H for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M190">View MathML</a>.

Lemma 3.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a>satisfies the (PS)γ-condition inHfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M192">View MathML</a>.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M186">View MathML</a> be a (PS)γ-sequence in H for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M196">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M198">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M199">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M200">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M186">View MathML</a> is bounded in H. Hence, there exist a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M203">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M46">View MathML</a> such that

Moreover, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M206">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M169">View MathML</a>. We use the Brézis-Lieb lemma to obtain (3.1) and (3.2) as follows:

(3.1)

(3.2)

Next, we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M210">View MathML</a>

(3.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M211">View MathML</a>

(3.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M212">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M213">View MathML</a>, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M214">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M215">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M216">View MathML</a>. By the Hölder inequality and the Sobolev embedding theorem, we get

Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M218">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183">View MathML</a>. By (A1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M97">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M222">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M223">View MathML</a>

(3.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M224">View MathML</a>. By (3.1)-(3.4) and Lemma 2.1, we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M225">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M226">View MathML</a>

(3.6)

We may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M227">View MathML</a>

(3.7)

Recall that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M228">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M229">View MathML</a>, by (3.5), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M230">View MathML</a>

This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M231">View MathML</a>. By (3.6) and (3.7), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M232">View MathML</a>

which is a contradiction. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M233">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M234">View MathML</a> strongly in H. □

4 Existence of k solutions

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M70">View MathML</a> be the unique, radially symmetric and positive ground state solution of equation (E0) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M1">View MathML</a>. Recall the facts (or see Bahri-Li [10], Bahri-Lions [11], Gidas-Ni-Nirenberg [12] and Kwong [13]):

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M237">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M238">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M239">View MathML</a>;

(ii) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M27">View MathML</a>, there exist positive numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M243">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M244">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M245">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M246">View MathML</a>

By Lien-Tzeng-Wang [14], then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M247">View MathML</a>

(4.1)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M249">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M250">View MathML</a>.

First of all, we want to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M251">View MathML</a>

Lemma 4.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M252">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M253">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M254">View MathML</a>

Moreover, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M255">View MathML</a>

Proof Part I: Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a> is continuous in H, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M257">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M258">View MathML</a> is uniformly bounded in H for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M27">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82">View MathML</a>, then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M261">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M262">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M27">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M264">View MathML</a>

From (A1), we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M265">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M266">View MathML</a>

It follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M267">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M268">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M27">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M270">View MathML</a>

From now on, we only need to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M271">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M272">View MathML</a>

and by (4.1), then

(4.2)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M274">View MathML</a>, by (4.2), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M275">View MathML</a>

Since

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M277">View MathML</a>

that is, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M253">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M280">View MathML</a>

Part II: By Lemma 2.4, there is a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M281">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M282">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82">View MathML</a>. Hence, from the result of Part I, we have that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M253">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M286">View MathML</a>

 □

Proof of Theorem 1.1 By Lemma 3.2, there exists a (PS)-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M288">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M54">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M291">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M253">View MathML</a>, by Lemma 3.3, there exist a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M288">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M295">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M296">View MathML</a> strongly in H. It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M84">View MathML</a> is a nontrivial solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M176">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M300">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M301">View MathML</a>, by Lemma 2.6, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M303">View MathML</a>. Applying the maximum principle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M304">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M305">View MathML</a> in Ω. □

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M306">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M307">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M308">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M309">View MathML</a>, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M310">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M311">View MathML</a>. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M312">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M313">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M314">View MathML</a> be given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M315">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M316">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M317">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M318">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M319">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M320">View MathML</a>.

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82">View MathML</a>, we define

By Lemma 2.4, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M281">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M324">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82">View MathML</a>. Then we have the following result.

Lemma 4.2There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M326">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M327">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M328">View MathML</a>for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82">View MathML</a>.

Proof Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M330">View MathML</a>

there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M326">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M332">View MathML</a>

 □

We need the following lemmas to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M333">View MathML</a> for sufficiently small ε, λ, μ.

Lemma 4.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M334">View MathML</a>.

Proof From Part I of Lemma 4.1, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M335">View MathML</a> uniformly in i. Similarly to Lemma 2.4, there is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M336">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M337">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M338">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M339">View MathML</a> be a minimizing sequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M340">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M341">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M342">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M343">View MathML</a>

We may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M344">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M345">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M347">View MathML</a>. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M348">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M349">View MathML</a>. We can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M350">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M351">View MathML</a>. □

Lemma 4.4There exists a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M352">View MathML</a>such that if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M353">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M354">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M355">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M356">View MathML</a>.

Proof On the contrary, there exist the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M357">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M358">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M359">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M360">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M362">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M363">View MathML</a>. It is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M186">View MathML</a> is bounded in H. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M365">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M366">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M367">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M368">View MathML</a>

which is a contradiction. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M369">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M183">View MathML</a>. Similarly to the concentration-compactness principle (see Lions [15,16] or Wang [[6], Lemma 2.16]), then there exist a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M371">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M372">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M373">View MathML</a>

(4.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M374">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M375">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M376">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M377">View MathML</a>. Then there are a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M378">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M379">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M380">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M381">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M382">View MathML</a>. Using the similar computation of Lemma 2.4, there is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M383">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M384">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M385">View MathML</a>

We deduce that a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M386">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M387">View MathML</a>. Then there are a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M388">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M389">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M390">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M391">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M382">View MathML</a>. By (4.3), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M393">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M394">View MathML</a>. Applying Ekeland’s variational principle, there exists a (PS)-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M396">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M341">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M398">View MathML</a>. Similarly to the proof of Lemma 3.3, there exist a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M399">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M400">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M401">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M402">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M382">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M404">View MathML</a>. Now, we want to show that there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M405">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M406">View MathML</a>.

(i) Claim that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M407">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M1">View MathML</a>. On the contrary, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M409">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M410">View MathML</a>

which is a contradiction.

(ii) Claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M411">View MathML</a> On the contrary, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M412">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M413">View MathML</a>. Then use the argument of (i) to obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M414">View MathML</a>

which is a contradiction.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M415">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M401">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M402">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M418">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M419">View MathML</a>

which is a contradiction.

Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M352">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M421">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M422">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M423">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M424">View MathML</a>. □

Lemma 4.5If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M425">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M426">View MathML</a>, then there exists a number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M427">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M428">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M424">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M90">View MathML</a>.

Proof Using the similar computation in Lemma 2.4, we obtain that there is the unique positive number

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M431">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M432">View MathML</a>. We want to show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M433">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M434">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M435">View MathML</a> for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M436">View MathML</a> (independent of u and v). First, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M437">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M438">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M439">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M440">View MathML</a>

(4.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M441">View MathML</a>

(4.5)

Moreover, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M442">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M120">View MathML</a>. It follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M433">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M445">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M446">View MathML</a>

(4.6)

Hence, by (4.4), (4.5) and (4.6), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M435">View MathML</a> for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M436">View MathML</a> (independent of u and v) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M434">View MathML</a>. Now, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M450">View MathML</a>

From the above inequality, we deduce that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M356">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M434">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M453">View MathML</a>

Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M454">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M90">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M456">View MathML</a>

By Lemma 4.4, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M457">View MathML</a>

or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M458">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M459">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M90">View MathML</a>. □

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M20">View MathML</a>, then by Lemma 4.3,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M462">View MathML</a>

(4.7)

By Lemmas 4.1, 4.2 and (4.7), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M89">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M464">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M465">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M466">View MathML</a>

(4.8)

Applying above Lemma 4.5, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M467">View MathML</a>

(4.9)

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82">View MathML</a>, by (4.8) and (4.9), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M469">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M470">View MathML</a>

Then applying Ekeland’s variational principle and using the standard computation, we have the following lemma.

Lemma 4.6For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82">View MathML</a>, there is a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M472">View MathML</a>-sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M473">View MathML</a>inHfor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M48">View MathML</a>.

Proof See Cao-Zhou [8]. □

Proof of Theorem 1.2 For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M90">View MathML</a>, by Lemma 4.6, there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M477">View MathML</a>-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M478">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M82">View MathML</a>. By (4.8), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M481">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a> satisfies the (PS)γ-condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M483">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M48">View MathML</a> has at least k critical points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M485">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M487">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M488">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M489">View MathML</a>. Replace the terms <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M490">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M491">View MathML</a> of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M43">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M493">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M494">View MathML</a>, respectively. It follows that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>) has k nonnegative solutions. Applying the maximum principle, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/118/mathml/M35">View MathML</a>) admits at least k positive solutions. □

Competing interests

The author declares that he has no competing interests.

Acknowledgements

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

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