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Multiple solutions for a fourth-order nonlinear elliptic problem which is superlinear at +∞ and linear at −∞

Ruichang Pei12* and Jihui Zhang2

Author Affiliations

1 School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, P.R. China

2 School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing, 210097, P.R. China

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Boundary Value Problems 2014, 2014:12  doi:10.1186/1687-2770-2014-12


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/12


Received:14 August 2013
Accepted:16 December 2013
Published:10 January 2014

© 2014 Pei and Zhang; 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

We consider a semilinear fourth-order elliptic equation with a right-hand side nonlinearity which exhibits an asymmetric growth at +∞ and at −∞. Namely, it is linear at −∞ and superlinear at +∞. Combining variational methods with Morse theory, we show that the problem has at least two nontrivial solutions, one of which is negative.

Keywords:
fourth-order elliptic boundary value problems; multiple solutions; critical groups; Morse theory

1 Introduction

Consider the following Navier boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M2">View MathML</a> is the biharmonic operator, and Ω is a bounded smooth domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M3">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M4">View MathML</a>), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M5">View MathML</a> the first eigenvalue of −△ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M6">View MathML</a>.

The conditions imposed on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7">View MathML</a> are as follows:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M9">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M11">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M12">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10">View MathML</a>;

(H2) there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M15">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M16">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M18">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M19">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M4">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M21">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10">View MathML</a>, where l is a nonnegative constant;

(H4) there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M23">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M24">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M25">View MathML</a>

(2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M26">View MathML</a>

(3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M27">View MathML</a>

(4)

(H5) there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M28">View MathML</a> and an integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M29">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M30">View MathML</a>

(5)

the first and the last inequality are strict on sets (not necessary the same) of positive measure, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M31">View MathML</a>

(6)

In view of the conditions (H3) and equation (3) in (H4), it is clear that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7">View MathML</a> is linear at −∞ and superlinear at +∞. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M34">View MathML</a> is a trivial solution of problem (1). It follows from (H1) and (H2) that the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M35">View MathML</a>

(7)

is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M36">View MathML</a> on the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M37">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M38">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M24">View MathML</a>. Under the condition (H2), the critical points of I are solutions of problem (1). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M40">View MathML</a> be the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M42">View MathML</a> be the eigenfunction corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M43">View MathML</a>. In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M44">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M45">View MathML</a> denote the eigenspace associated with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M46">View MathML</a>. Throughout this article, we denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M47">View MathML</a> the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M48">View MathML</a> norm and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M49">View MathML</a>. The aim of this paper is to prove a multiplicity theorem for problem (1) when the nonlinearity term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7">View MathML</a> exhibits an asymmetric behavior as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M51">View MathML</a> approaches +∞ and −∞. In the past, some authors studied the following elliptic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M52">View MathML</a>

(8)

with asymmetric nonlinearities by using the Fučík spectrum of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M53">View MathML</a>. This approach requires that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7">View MathML</a> exhibits linear growth at both +∞ and −∞ and that the limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M55">View MathML</a> exist and belong to ℝ. See the works of Các [1], Dancer and Zhang [2], Magalhães [3], de Paiva [4], Schechter [5] and the references therein. Equations with nonlinearities which are superlinear in one direction and linear in the other were investigated by Arcoya and Villegas [6] and Perera [7]. They let the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7">View MathML</a> be line at −∞ and satisfy the Ambrosetti-Rabinowitz condition at +∞. Particularly, it is worth noticing paper [8]. The authors relax several of the above restrictions on the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7">View MathML</a>. Their nonlinearity is only measurable in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10">View MathML</a>. The limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M59">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M60">View MathML</a> need not exist and the growth at −∞ can be linear or sublinear. Furthermore, their nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7">View MathML</a> does not satisfy the famous AR-condition. They use the truncated skill of first order weak derivative to verify the (PS) condition and obtain multiple solutions for problem (1) by combining variational methods and Morse theory.

To the authors’ knowledge, there seem to be few results on problem (1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M7">View MathML</a> is asymmetric nonlinearity at positive infinity and at negative infinity. However, the method in [8] cannot be applied directly to the biharmonic problems. For example, for the Laplacian problem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M63">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M64">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M66">View MathML</a>. We can use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M67">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M68">View MathML</a> as a test function, which is helpful in proving a solution nonnegative. While for the biharmonic problems, this trick fails completely since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M69">View MathML</a> does not imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M70">View MathML</a> (see [[9], Remark 2.1.10] and [10,11]). As far as this point is concerned, we will make use of the new methods to overcome it.

This fourth-order semilinear elliptic problem can be considered as an analogue of a class of second-order problems which have been studied by many authors. In [12], there was a survey of results obtained in this direction. In [13], Micheletti and Pistoia showed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M71">View MathML</a> admits at least two solutions by a variation of linking if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M72">View MathML</a> is sublinear. Chipot [14] proved that the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M71">View MathML</a> has at least three solutions by a variational reduction method and a degree argument. In [15], Zhang and Li showed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M71">View MathML</a> admits at least two nontrivial solutions by Morse theory and local linking if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M72">View MathML</a> is superlinear and subcritical on u.

In this article, under the guidance of [8], we consider multiple solutions of problem (1) with the asymmetric nonlinearity by using variational methods and Morse theory.

2 Main result and auxiliary lemmas

Let us now state the main result.

Theorem 2.1Assume conditions (H1)-(H5) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M76">View MathML</a>, then problem (1) has at least two nontrivial solutions.

Lemma 2.2Under the assumptions of Theorem 2.1, thenIsatisfies the (PS) condition.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M77">View MathML</a> be a sequence such that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M78">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M79">View MathML</a>

(9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M80">View MathML</a>

(10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M81">View MathML</a> is a positive constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M82">View MathML</a> is a sequence which converges to zero. By a standard argument, in order to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M83">View MathML</a> has a convergence subsequence, we have to show that it is a bounded sequence. To do this, we argue by contradiction assuming that for a subsequence, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M84">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M85">View MathML</a>

Without loss of generality we can assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M86">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M87">View MathML</a> and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M88">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M89">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M90">View MathML</a> and then it is possible to extract a subsequence (denoted also by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M91">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M92">View MathML</a>

(11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M93">View MathML</a>

(12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M94">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M95">View MathML</a>

(14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M97">View MathML</a>. Dividing both sides of inequality (10) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M98">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M99">View MathML</a>

Passing to the limit we deduce from equation (11) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M100">View MathML</a>

(15)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M101">View MathML</a>.

Now we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M102">View MathML</a> a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10">View MathML</a>. To verify this, let us observe that by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M104">View MathML</a> in equation (15) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M105">View MathML</a>

(16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M106">View MathML</a>. But, on the other hand, from (H3) and equation (3) in (H4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M107">View MathML</a>

for some positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M108">View MathML</a>. Moreover, using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M109">View MathML</a> a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M110">View MathML</a>, equation (13) and the superlinearity of f, we also deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M111">View MathML</a>

Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M112">View MathML</a> we will obtain by Fatou’s lemma that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M113">View MathML</a>

which contradicts inequality (16). Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M114">View MathML</a> and the claim is proved.

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M115">View MathML</a>, by (H3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M116">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M117">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10">View MathML</a>. By using Lebesgue dominated convergence theorem in equation (15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M119">View MathML</a>

(17)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M101">View MathML</a>. This contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M121">View MathML</a>. □

Lemma 2.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M122">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M123">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M124">View MathML</a>is an integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M126">View MathML</a>a.e. on Ω and the inequality is strict on a set of positive measure, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M127">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M128">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M129">View MathML</a>.

Proof We claim that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M130">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M131">View MathML</a>

(18)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M129">View MathML</a>. In fact, if not, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M84">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M134">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M78">View MathML</a>, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M136">View MathML</a> for all n. By the homogeneity of the above inequality, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M137">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M138">View MathML</a>

(19)

for all n. It follows from the weak compactness of the unit ball of W that there exists a subsequence, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M84">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M140">View MathML</a> weakly converges to u in W. Now Sobolev’s embedding theorem suggests that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M84">View MathML</a> converges to u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M142">View MathML</a>. From inequality (19) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M143">View MathML</a>

Moreover one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M144">View MathML</a>

Hence we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M145">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M146">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M34">View MathML</a> on a positive measure subset. It contradicts the unique continuation property of the eigenfunction. □

3 Computation of the critical groups

It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equation. Let us recall some results which will be used later. We refer the readers to the book [16] for more information on Morse theory.

Let H be a Hilbert space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M149">View MathML</a> be a functional satisfying the (PS) condition or (C) condition, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M150">View MathML</a> be the qth singular relative homology group with integer coefficients. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M151">View MathML</a> be an isolated critical point of I with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M153">View MathML</a>, and U be a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M151">View MathML</a>. The group

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M155">View MathML</a>

is said to be the qth critical group of I at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M151">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M157">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M158">View MathML</a> be the set of critical points of I and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M159">View MathML</a>, the critical groups of I at infinity are formally defined by [17]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M160">View MathML</a>

From the deformation theorem, we see that the above definition is independent of the particular choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M161">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M161">View MathML</a> then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M163">View MathML</a>

(20)

For the convenience of our proof, we first recall two interesting results and prove two important propositions.

Proposition 3.1[18]

Under (H2), if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M164">View MathML</a>is an isolated critical point ofI, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M165">View MathML</a>.

Proposition 3.2[19]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M166">View MathML</a>and for some integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M124">View MathML</a>we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M168">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M169">View MathML</a>, then either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M170">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M171">View MathML</a>.

Proposition 3.3If the assumptions of Theorem 2.1 hold, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M172">View MathML</a>

Proof Under the guidance of [8] and [18], we begin to prove this result. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M173">View MathML</a>. Indeed, it follows from above Proposition 3.1 that I and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M174">View MathML</a> have same critical set. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M175">View MathML</a> is dense in E, invoking Proposition 16 of Palais [20], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M176">View MathML</a>

(21)

From equations (20) and (21), we see that in order to prove the proposition, it suffices to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M177">View MathML</a>

(22)

In order to prove equation (22), we proceed as follows. We define the sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M178">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M179">View MathML</a>

Consider the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M180">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M181">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M182">View MathML</a> is a continuous homotopy and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M183">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M184">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M185">View MathML</a> is contractible in itself.

By equation (3) in (H4), given any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M127">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M187">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M188">View MathML</a>

(23)

Similarly, from condition (H3), and by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M116">View MathML</a> even bigger if necessary, we observe that there is a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M190">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M191">View MathML</a>

(24)

Moreover, by condition (H2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M192">View MathML</a>

(25)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M193">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M194">View MathML</a>. By inequalities (23), (24), and (25), for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M195">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M196">View MathML</a>

(26)

Recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M197">View MathML</a> is arbitrary, from (26), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M198">View MathML</a>

(27)

Using formula (4) in condition (H4), we see that there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M199">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M200">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M201">View MathML</a>

(28)

By (H2) and formula (2) in condition (H4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M202">View MathML</a>

(29)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M116">View MathML</a>. By inequalities (28) and (29), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M204">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M205">View MathML</a>

(30)

where C is a positive constant. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M206">View MathML</a> be the continuous embedding map. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M207">View MathML</a> denote the duality brackets for the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M208">View MathML</a>. We let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M209">View MathML</a>, and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M210">View MathML</a>

(31)

Then, from equation (27), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M211">View MathML</a>

(32)

From conditions (H2) and (H3), we see that given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M212">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M200">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M214">View MathML</a>

(33)

Using inequality (33), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M215">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M216">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M217">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M218">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M116">View MathML</a> is a positive constant. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M220">View MathML</a> is coercive, thus we find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M221">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M222">View MathML</a>. We pick

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M223">View MathML</a>

Then inequality (32) implies that we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M224">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M225">View MathML</a>

Moreover, the implicit function theorem implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M226">View MathML</a>.

By the choice of a, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M227">View MathML</a>

(34)

We define the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M228">View MathML</a>. The map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M229">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M230">View MathML</a>

(35)

is a continuous deformation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M232">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M233">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M234">View MathML</a> (see equations (34) and (35)). Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M235">View MathML</a> is a strong deformation retract of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M231">View MathML</a>. Hence we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M237">View MathML</a>

(36)

Recalling that in the first part of the proof, we established that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M185">View MathML</a> is contractible. This yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M239">View MathML</a>

Combining with equation (36) leads to equation (22), which completes the proof. □

Proposition 3.4If the assumptions of Theorem 2.1 hold, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M240">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M241">View MathML</a> (Vbeing defined in Lemma 2.3).

Proof By condition (H5), given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M212">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M243">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M244">View MathML</a>

(37)

Since V is finite dimensional, all norms are equivalent. Thus we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M245">View MathML</a> small such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M246">View MathML</a>

(38)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M247">View MathML</a>. Taking inequalities (37) and (38) into account, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M247">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M249">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M250">View MathML</a>

(39)

Similar to the proof of Lemma 2.3, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M116">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M252">View MathML</a>

(40)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M247">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M249">View MathML</a>.

On the other hand, for given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M212">View MathML</a>, it follows from (H2) and (H5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M256">View MathML</a>

(41)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M258">View MathML</a>. By (41) and Lemma 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M259">View MathML</a>

(42)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M129">View MathML</a>. From inequality (42), we infer that for ρ small enough we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M261">View MathML</a>

(43)

From inequalities (40) and (43), we know that I has a local linking at 0. Then invoking Proposition 2.3 of Bartsch and Li [17], we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M262">View MathML</a>. □

4 Proof of the main result

Proof of Theorem 2.1 We consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M263">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M264">View MathML</a>

Define a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M265">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M266">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M267">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M268">View MathML</a>. Obviously, by conditions (H1) and (H3), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M269">View MathML</a> is coercive and boundedness from below. Thus we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M270">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M271">View MathML</a>

(44)

Next, we claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M272">View MathML</a>. By condition (H5), given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M273">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M274">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M275">View MathML</a>

(45)

For s small enough, it follows from inequality (45) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M276">View MathML</a>

and thus, by equation (44), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M277">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M278">View MathML</a>. From condition (H1) and strong maximum principle, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M279">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M280">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M281">View MathML</a> is an interior point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M282">View MathML</a>, from Proposition 3.1, we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M283">View MathML</a>

(46)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M284">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M212">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M286">View MathML</a>. We consider the sublevel sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M287">View MathML</a>

Suppose that 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M281">View MathML</a> are the only critical points of I. Otherwise, we have a second nontrivial smooth solution and so we are done. By Proposition 3.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M289">View MathML</a>

(47)

We know that I satisfies the (PS) condition (see Lemma 2.2). Hence choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M212">View MathML</a> small enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M291">View MathML</a>

(48)

(see Proposition 3.4). Because of equations (47) and (48), using Proposition 3.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M292">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M293">View MathML</a>, then there is a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M294">View MathML</a> of I such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M295">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M296">View MathML</a>, then there is a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M294">View MathML</a> of I such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M298">View MathML</a>

(49)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M299">View MathML</a>, from equations (46) and (49), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M300">View MathML</a>. It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M301">View MathML</a>. Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M281">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/12/mathml/M303">View MathML</a> are two solutions of problem (1). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions in improving this article. This study was supported by the National NSF (Grant No. 11101319) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No. 1301038C).

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