Open Access Research

Superlinear gradient system with a parameter

Anran Li* and Jiabao Su

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School of Mathematical Sciences, Capital Normal University, Beijing, 100048, People’s Republic of China

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Citation and License

Boundary Value Problems 2012, 2012:110  doi:10.1186/1687-2770-2012-110


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/110


Received:30 July 2012
Accepted:25 September 2012
Published:9 October 2012

© 2012 Li and Su; 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 study the multiplicity of nontrivial solutions for a superlinear gradient system with saddle structure at the origin. We make use of a combination of bifurcation theory, topological linking and Morse theory.

MSC: 35J10, 35J65, 58E05.

Keywords:
gradient system; superlinear; critical group; Morse theory; linking

1 Introduction

In this paper, we study the existence of multiple solutions to the gradient system

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M2">View MathML</a> is a bounded open domain with a smooth boundary Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M3">View MathML</a>, λ is a real parameter and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M4">View MathML</a> is fixed. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M5">View MathML</a> is the set of all continuous, cooperative and symmetric matrix functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M6">View MathML</a>. A matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M7">View MathML</a> takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M8">View MathML</a>

with the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M9">View MathML</a> satisfying the conditions that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M10">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M11">View MathML</a>, which means A is cooperative and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M12">View MathML</a>.

We impose the following assumptions on the function F:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M14">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M18">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20">View MathML</a>) There is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M22">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M23">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>) There is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M26">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M27">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M29">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M31">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M32">View MathML</a> small and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M35">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M32">View MathML</a> small and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19">View MathML</a>.

Here and in the sequel, 0 is used to denote the origin in various spaces, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M39">View MathML</a> denote the norm and the inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M6">View MathML</a>, Bz denotes the matrix product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M41">View MathML</a> for a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M42">View MathML</a> matrix B and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M43">View MathML</a>. For two symmetric matrices B and C in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M45">View MathML</a> means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M46">View MathML</a> is positive definite.

Let E be the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M47">View MathML</a> endowed with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M48">View MathML</a>

and the associated norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M49">View MathML</a>

By the compact Sobolev embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M50">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M51">View MathML</a>, under the assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20">View MathML</a>), the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M54">View MathML</a>

(1.1)

is well defined and is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M55">View MathML</a> (see [1]) with derivatives

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M60">View MathML</a>. Therefore, the solutions to (GS)λ are exactly critical points of Φ in E.

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15">View MathML</a>) the system (GS)λ admits a trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62">View MathML</a> for any fixed parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M63">View MathML</a>. We are interested in finding nontrivial solutions to (GS)λ. The existence of nontrivial solutions of (GS)λ depends on the behaviors of F near zero and infinity. The purpose of this paper is to find multiple nontrivial solutions to (GS)λ with superlinear term when the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62">View MathML</a> acts as a local saddle point of the energy functional Φ in the sense that the parameter λ is close to a higher eigenvalue of the linear gradient system with the given weight matrix A

It is known (see [2,3]) that for a given matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M66">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67">View MathML</a>) admits a sequence of distinct eigenvalues of finite multiplicity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M68">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M69">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M70">View MathML</a>.

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M71">View MathML</a> the negative part of F, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M72">View MathML</a>.

We will prove the following theorems.

Theorem 1.1Assume (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30">View MathML</a>) and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M76">View MathML</a>be fixed. Then there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77">View MathML</a>such that when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M78">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M79">View MathML</a>, (GS)λ has at least three nontrivial solutions inE.

Theorem 1.2Assume (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34">View MathML</a>) and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M76">View MathML</a>be fixed. Then there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77">View MathML</a>such that when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M78">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M86">View MathML</a>, (GS)λ has at least three nontrivial solutions inE.

Theorem 1.3Assume (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M89">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M91">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M92">View MathML</a>small. Then there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77">View MathML</a>such that when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M78">View MathML</a>, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M95">View MathML</a>, (GS)λ has at least two nontrivial solutions inE.

We give some comments and comparisons. The superlinear problems have been studied extensively via variational methods since the pioneering work of Ambrosetti and Rabinowitz [4]. Most known results on elliptic superlinear problems are contributed to a single equation with Dirichlet boundary data. Let us mention some historical progress on a single equation. When the trivial solution 0 acted as a local minimizer of the energy functional, one positive solution and one negative solution were obtained by using the mountain-pass theorem in [4] and the cut-off techniques; and a third solution was constructed in a famous paper of Wang [5] by using a two dimensional linking method and a Morse theoretic approach. When the trivial solution 0 acted as a local saddle point of the energy functional, the existence of one nontrivial solution was obtained by applying a critical point theorem, which is now well known as the generalized mountain-pass theorem, built by Rabinowitz in [6] under a global sign condition (see [1]). Some extensions were done in [7,8]via local linking. More recently, in the work of Rabinowitz, Su and Wang [9], multiple solutions have been obtained by combining bifurcation methods, Morse theory and homological linking when 0 is a saddle point in the sense that the parameter λ is very close to a higher eigenvalue of the related linear operator.

In the current paper, we build multiplicity results for superlinear gradient systems by applying the ideas constructed in [9]. These results are new since, to the best of our knowledge, no multiplicity results for gradient systems have appeared in the literature for the case that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62">View MathML</a> is a saddle point of Φ.

We give some explanations regarding the conditions and conclusions. The assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>) are standard in the study of superlinear problems. (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34">View MathML</a>) are used for bifurcation analysis. It sees that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30">View MathML</a>) implies that F is positive near zero, while (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34">View MathML</a>) implies that F must be negative near zero. The local properties of F near zero are necessary for constructing homological linking. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M103">View MathML</a>, for any parameter λ in a bounded interval, say in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M104">View MathML</a>, one can use the same arguments as in [1] to construct linking starting from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105">View MathML</a>. In our theorems, we do not require the global sign condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M106">View MathML</a>. When the parameter λ is close to the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105">View MathML</a>, the homological linking will be constructed starting from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M108">View MathML</a> and this linking is different from the one in [1]. This reveals the fact that when λ is close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M109">View MathML</a> from the right-hand side, the linking starting from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M108">View MathML</a> can still be constructed even if F is negative somewhere. The conditions similar to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34">View MathML</a>) were first introduced in [10] where multiple periodic solutions for the second-order Hamiltonian systems were studied via the ideas in [9]. Since we treat a different problem in the current paper, we need to present the detailed discussions although some arguments may be similar to those in [9,10].

The paper is organized as follows. In Section 2, we collect some basic abstract tools. In Section 3, we get solutions by linking arguments and give partial estimates of homological information. In Section 4, we get solutions by bifurcation theorem and give the estimates of the Morse index. The final proofs of Theorems 1.1-1.3 are given in Section 5.

2 Preliminary

In this section, we give some preliminaries that will be used to prove the main results of the paper. We first collect some basic results on the Morse theory for a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M55">View MathML</a> functional defined on a Hilbert space.

Let E be a Hilbert space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M114">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M115">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M117">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M118">View MathML</a>. We say that Φ satisfies the (PS)c condition at the level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M119">View MathML</a> if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M120">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M122">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M123">View MathML</a>, has a convergent subsequence. Φ satisfies (PS) if Φ satisfies (PS)c at any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M118">View MathML</a>.

We assume that Φ satisfies (PS) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M125">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M126">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M127">View MathML</a> and U be a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M128">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M129">View MathML</a>. The group

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M130">View MathML</a>

is called the qth critical group of Φ at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M128">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M132">View MathML</a> denotes a singular relative homology group of the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M133">View MathML</a> with coefficients field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M134">View MathML</a> (see [11,12]).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M135">View MathML</a>. The group

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M136">View MathML</a>

is called the qth critical group of Φ at infinity (see [13]).

We call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M137">View MathML</a> the qth Morse-type numbers of the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M138">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M139">View MathML</a> the Betti numbers of the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M140">View MathML</a>. The core of the Morse theory [11,12] is the following relations between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M142">View MathML</a>:

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M144">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M145">View MathML</a> for all q. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M146">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M147">View MathML</a>, it follows that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M148">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M149">View MathML</a>, then Φ must have a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M150">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M151">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M152">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M153">View MathML</a> for all q. Thus, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M154">View MathML</a> for some q, then Φ must have a new critical point. One can use critical groups to distinguish critical points obtained by other methods and use the Morse equality to find new critical points.

For the critical groups of Φ at an isolated critical point, we have the following basic facts (see [11,12]).

Proposition 2.1Assume thatzis an isolated critical point of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M114">View MathML</a>with a finite Morse index<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M156">View MathML</a>and nullity<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M157">View MathML</a>. Then

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M158">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M159">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M160">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M161">View MathML</a> (Gromoll-Meyer [14]);

(3) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M162">View MathML</a>then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M163">View MathML</a>;

(4) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M164">View MathML</a>then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M165">View MathML</a>.

Proposition 2.2 ([15,16])

Let 0 be an isolated critical point of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M114">View MathML</a>with a finite Morse index<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M167">View MathML</a>and nullity<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M168">View MathML</a>. Assume that Φ has a local linking at 0 with respect to a direct sum decomposition<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M170">View MathML</a>, i.e., there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M171">View MathML</a>small such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M172">View MathML</a>

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M173">View MathML</a>for either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M174">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M175">View MathML</a>.

The concept of local linking was introduced in [7]. In [15] a partial result was given for a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M176">View MathML</a> functional. The above result was obtained in [16].

Now, we recall an abstract linking theorem which is from [1,12,15].

Proposition 2.3 ([1,12,15])

LetEbe a real Banach space with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M177">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M178">View MathML</a>be finite. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M179">View MathML</a>satisfies (PS) and

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M180">View MathML</a>) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M181">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M182">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M183">View MathML</a>

(2.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M184">View MathML</a>,

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M185">View MathML</a>) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M186">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M187">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M188">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M189">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M190">View MathML</a>

Then Φ has a critical point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M150">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M192">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M193">View MathML</a>

(2.3)

We note here that under the framework of Proposition 2.3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M194">View MathML</a> and ∂Qhomotopically link with respect to the direct sum decomposition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M195">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M194">View MathML</a> and ∂Q are also homologically linked. The conclusion (2.3) follows from Theorems 1.1′ and 1.5 of Chapter II in [12]. (See also [15].)

We finally collect some properties of the eigenvalue problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67">View MathML</a>). Associated with a matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M66">View MathML</a>, there is a compact self-adjoint operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M199">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M200">View MathML</a>

The compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M201">View MathML</a> follows from the compact embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M202">View MathML</a>. The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M201">View MathML</a> possesses the property that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M204">View MathML</a> is an eigenvalue of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67">View MathML</a>) if and only if there is nonzero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M206">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M207">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67">View MathML</a>) has the sequence of distinct eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M209">View MathML</a>

and each eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M204">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67">View MathML</a>) has a finite multiplicity. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M212">View MathML</a>, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M213">View MathML</a>

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M214">View MathML</a>

Then the following variational inequalities hold:

We refer to [2,3] for more properties related to the eigenvalue problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67">View MathML</a>) and the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M201">View MathML</a>.

3 Solutions via homological linking

In this section, we give the existence a nontrivial solution of (GS)λ by applying homological linking arguments and then give some estimate of its Morse index. The following lemmas are needed.

Lemma 3.1Assume thatFsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>), then for any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M63">View MathML</a>, the functional Φ satisfies the (PS) condition.

Proof By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20">View MathML</a>) and the compact embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M222">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M223">View MathML</a>, it is enough to show that any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M224">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M225">View MathML</a>

(3.1)

is bounded in E. Here and below, we use C to denote various positive constants. We modify the arguments in [1]. Choosing a positive number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M226">View MathML</a> for n large, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M227">View MathML</a>

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>) we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M229">View MathML</a>

(3.2)

Therefore,

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M231">View MathML</a>. By the Hölder inequality and the Young inequality, we get for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M232">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M233">View MathML</a>

(3.3)

Thus, for a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M232">View MathML</a> small enough, we have by (3.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M235">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M236">View MathML</a> is bounded in E. The proof is complete. □

Now, we construct a homological linking with respect to the direct sum decomposition of E for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M237">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M238">View MathML</a>

Take an eigenvector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M239">View MathML</a> corresponding to the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M108">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67">View MathML</a>) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M242">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M243">View MathML</a>

Lemma 3.2Assume thatFsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M76">View MathML</a>. Then there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M247">View MathML</a>small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M182">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M249">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M250">View MathML</a>

(3.4)

Proof By the conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20">View MathML</a>), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M232">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M254">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M255">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M256">View MathML</a> is the constant for the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M257">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M258">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M206">View MathML</a>. Since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M260">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M261">View MathML</a>

it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M262">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M263">View MathML</a> is independent of λ and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M264">View MathML</a>

(3.5)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M265">View MathML</a> and the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M266">View MathML</a> achieves its maximum

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M267">View MathML</a>

(3.6)

on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M268">View MathML</a> at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M269">View MathML</a>

(3.7)

we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M270">View MathML</a>

(3.8)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M271">View MathML</a> is a decreasing function with respect to λ for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M232">View MathML</a> small, (3.4) holds for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M273">View MathML</a>

The constants α and ρ are independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M274">View MathML</a>. The proof is complete. □

Lemma 3.3Assume thatFsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M76">View MathML</a>. Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M280">View MathML</a>, all independent ofλ, such that when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M281">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M282">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M283">View MathML</a>

(3.9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M284">View MathML</a>

Proof From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>) we deduce (3.2) with a positive constant C independent of λ. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M286">View MathML</a>, write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M287">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M290">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M291">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M292">View MathML</a>

(3.10)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M25">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M294">View MathML</a>, (3.10) shows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M278">View MathML</a> independent of λ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M296">View MathML</a>

(3.11)

Now, fix such an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M278">View MathML</a> that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M298">View MathML</a> with ρ given in Lemma 3.2. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M299">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M300">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M301">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M302">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M303">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M304">View MathML</a>. Then we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M305">View MathML</a>

(3.12)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M306">View MathML</a>, taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M307">View MathML</a>

then, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M281">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M309">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M310">View MathML</a>

The proof is complete. □

Now, we apply Proposition 2.3 to get the following existence result with partial homological information.

Theorem 3.4LetFsatisfy (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M313">View MathML</a>. Then there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77">View MathML</a>such that when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M309">View MathML</a>, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M281">View MathML</a>, (GS)λ has one nontrivial solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M317">View MathML</a>with a critical group satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M318">View MathML</a>

(3.13)

Proof By Lemma 3.1, Φ satisfies (PS). By Lemmas 3.2 and 3.3, for each fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M319">View MathML</a>, Φ satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M180">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M185">View MathML</a>) in the sense that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M322">View MathML</a>

(3.14)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M323">View MathML</a> and ∂Q homotopically link with respect to the decomposition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M324">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M325">View MathML</a>, it follows from Proposition 2.3 that Φ has a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M326">View MathML</a> with positive energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M327">View MathML</a> and its critical group satisfying (3.13). The proof is complete. □

We give some remarks. The existence of one nontrivial solution in Theorem 3.4 is valid when F is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M176">View MathML</a>. From Lemma 3.2, one sees that the energy of the obtained solution is bounded away from 0 as λ is close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105">View MathML</a>. A rough local sign condition on F is needed. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M103">View MathML</a>, then for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M331">View MathML</a>, a linking with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M332">View MathML</a> can be constructed. Proposition 2.3 is applied again to get a nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M150">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M334">View MathML</a>

(3.15)

Therefore, when a global sign condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M103">View MathML</a> is present, as λ is close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105">View MathML</a> from the left-hand side, two linkings can be constructed and two nontrivial solutions can be obtained. The question is how to distinguish <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M317">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M150">View MathML</a>. Theorem 3.4 includes the case that for λ close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105">View MathML</a> from the right-hand side, the linking with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M340">View MathML</a> is constructed provided the negative values of F are small. This phenomenon was first observed in [9].

4 Solutions via bifurcation

In this section, we get two solutions for (GS)λvia bifurcation arguments [1]. We first cite the bifurcation theorem in [1].

Proposition 4.1 (Theorem 11.35 in [1])

LetEbe a Hilbert space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M341">View MathML</a>with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M342">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M343">View MathML</a>is symmetric and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M344">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M345">View MathML</a>. Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M346">View MathML</a>

(4.1)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M347">View MathML</a>be an isolated eigenvalue of finite multiplicity. Then either

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M348">View MathML</a>is not an isolated solution of (4.1) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M349">View MathML</a>, or

(ii) there is a one-sided neighborhood Λ ofμsuch that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M350">View MathML</a>, (4.1) has at least two distinct nontrivial solutions, or

(iii) there is a neighborhood Λ ofμsuch that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M350">View MathML</a>, (4.1) has at least one nontrivial solution.

We apply Proposition 4.1 to get two nontrivial solutions of (GS)λ for λ close to an eigenvalue of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67">View MathML</a>) and then give the estimates of the Morse index.

Theorem 4.2Assume thatFsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20">View MathML</a>). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M76">View MathML</a>be fixed. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M356">View MathML</a>such that (GS)λ has at least two nontrivial solutions for

(1) every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M357">View MathML</a>if (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30">View MathML</a>) holds;

(2) every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M359">View MathML</a>if (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M34">View MathML</a>) holds.

Furthermore, the Morse index<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M361">View MathML</a>and the nullity<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M362">View MathML</a>of such a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M363">View MathML</a>satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M364">View MathML</a>

(4.2)

Proof Under the assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15">View MathML</a>), for each eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M367">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M369">View MathML</a> is a bifurcation point of (GS)λ (see [1]).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M370">View MathML</a> be a solution of (GS)λ near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M371">View MathML</a> which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M372">View MathML</a>

(4.3)

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M375">View MathML</a>

(4.4)

Let (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30">View MathML</a>) hold. By the elliptic regularity theory (see [17]), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M377">View MathML</a> small implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M378">View MathML</a> small. Then by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30">View MathML</a>), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M380">View MathML</a>

(4.5)

Now, consider the linear eigenvalue gradient system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M381">View MathML</a>

(4.6)

We denote the distinct eigenvalues of (4.6) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M382">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M383">View MathML</a>. By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15">View MathML</a>), if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62">View MathML</a>, then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M386">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M387">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M388">View MathML</a>. By (4.5), the standard variational characterization of the eigenvalues of (4.6) shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M389">View MathML</a> is less than the corresponding jth ordered eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M367">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M67">View MathML</a>). Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M392">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M393">View MathML</a> in E. By (4.3) and (4.4), we see that z is a solution of (4.6) with eigenvalue λ. It must be that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M394">View MathML</a> since λ is close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M105">View MathML</a>. Therefore, the case (ii) of Proposition 4.1 occurs under the given conditions. This proves the case (1). The existence for the case (2) is proved in the same way.

Now, we estimate the Morse indices for the solutions obtained above. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M363">View MathML</a> be a bifurcation solution of (GS)λ. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M397">View MathML</a>

Applying the elliptic regularity theory, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M398">View MathML</a>

(4.7)

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M399">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M400">View MathML</a>

Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M401">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M402">View MathML</a>

and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M403">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M404">View MathML</a>

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15">View MathML</a>) and (4.7), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M77">View MathML</a> such that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M407">View MathML</a>,

Therefore, the Morse index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M361">View MathML</a> and the nullity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M362">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M363">View MathML</a> satisfy (4.2). The proof is complete. □

5 Proofs of main theorems

In this section, we give the proof of main theorems in this paper. We first compute the critical groups of Φ at both infinity and zero.

Lemma 5.1 (see [5])

LetFsatisfy (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M24">View MathML</a>), then for any fixed<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M63">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M415">View MathML</a>

(5.1)

Proof The idea of the proof comes from the famous paper [5]. We include a sketched proof in an abstract version. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M63">View MathML</a>, denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M417">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M418">View MathML</a>. Modifying the arguments in [5], we get the following facts:

(5.2)

(5.3)

The following arguments are from [10]. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M421">View MathML</a>, it follows from (5.2) and (5.3) that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M422">View MathML</a>, there is a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M423">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M424">View MathML</a>

(5.4)

By (5.4) and the implicit function theorem, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M425">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M426">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M427">View MathML</a>. Define a map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M428">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M429">View MathML</a>

(5.5)

Clearly, ϱ is continuous, and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M430">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M431">View MathML</a>, by (5.4),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M432">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M433">View MathML</a>

and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M434">View MathML</a> is a strong deformation retract of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M435">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M436">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M437">View MathML</a> is contractible, which follows from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M438">View MathML</a>. □

Lemma 5.2LetFsatisfy (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M13">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20">View MathML</a>).

(1) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M441">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M442">View MathML</a>.

(2) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M443">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M444">View MathML</a>.

(3) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M445">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M446">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M92">View MathML</a>small, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M448">View MathML</a>.

(4) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M445">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M450">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M92">View MathML</a>small, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M452">View MathML</a>.

Proof By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M454">View MathML</a>

(1) When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M441">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62">View MathML</a> is a nondegenerate critical point of Φ with the Morse index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M457">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M458">View MathML</a>.

(2) When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M443">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62">View MathML</a> is a nondegenerate critical point of Φ with the Morse index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M461">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M462">View MathML</a>.

(3) When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M445">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62">View MathML</a> is a degenerate critical point of Φ with the Morse index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M465">View MathML</a> and the nullity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M466">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M467">View MathML</a>.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M446">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M469">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M470">View MathML</a> small. We will show that Φ has a local linking structure at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62">View MathML</a> with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M472">View MathML</a>. If this has been done, then by Proposition 2.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M473">View MathML</a>.

Now, Φ can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M474">View MathML</a>

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M15">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20">View MathML</a>), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M232">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M254">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M479">View MathML</a>

Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M480">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M481">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M482">View MathML</a> is finite dimensional, all norms on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M482">View MathML</a> are equivalent, hence for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M171">View MathML</a> small,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M485">View MathML</a>

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M20">View MathML</a>), we have that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M21">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M488">View MathML</a>

(5.6)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M489">View MathML</a>, we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M490">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M491">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M492">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M493">View MathML</a>

(5.7)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M495">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M496">View MathML</a>. Hence, by (5.6) and the Poincaré inequality, we have for various constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M21">View MathML</a>,

(5.8)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M500">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M501">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M502">View MathML</a>

(5.9)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M265">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M171">View MathML</a> small,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M505">View MathML</a>

(5.10)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M506">View MathML</a>, it must hold that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M507">View MathML</a>

(5.11)

Here we use a potential convention that (GS)λ has finitely many solutions and then 0 is isolated. Otherwise, one would have that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M171">View MathML</a> small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M509">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M510">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M512">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M19">View MathML</a>. Thus, 0 would not be an isolated critical point of Φ and (GS)λ would have infinitely many nontrivial solutions. By (5.10) and (5.11), we verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M514">View MathML</a>

Applying Proposition 2.2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M515">View MathML</a>

(4) When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M450">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M92">View MathML</a> small, a similar argument shows that Φ has a local linking structure at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62">View MathML</a> with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M519">View MathML</a>. By Proposition 2.2, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M520">View MathML</a>. □

Finally, we prove the theorems.

Proof of Theorem 1.1 It follows from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M30">View MathML</a>) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M450">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M32">View MathML</a> small. By Theorem 3.4 for the part <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M524">View MathML</a>, (GS)λ has a nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M525">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M526">View MathML</a>

(5.12)

By Theorem 4.2(1), (GS)λ has two nontrivial solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M527">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M528">View MathML</a>) with their Morse indices satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M529">View MathML</a>

From Proposition 2.1(2), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M530">View MathML</a>

(5.13)

From (5.12) and (5.13), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M531">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M528">View MathML</a>). The proof is complete. □

Proof of Theorem 1.2 With the same argument as above, it follows from Theorem 4.2(2) and Theorem 3.4 for the part <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M533">View MathML</a>. We omit the details. □

Proof of Theorem 1.3 By Theorem 3.4 for the part <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M534">View MathML</a>, (GS)λ has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M525">View MathML</a> with its energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M536">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M537">View MathML</a>

(5.14)

By Lemma 5.1 and Lemma 5.2(3), we have that

(5.15)

(5.16)

Assume that (GS)λ has only two solutions 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M525">View MathML</a>. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M541">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M542">View MathML</a>. Then by the deformation and excision properties of singular homology (see [12]), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M543">View MathML</a>

(5.17)

By (5.17), the long exact sequences for the topological triple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M544">View MathML</a> read as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M545">View MathML</a>

(5.18)

We deduce by (5.15) and (5.18) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M546">View MathML</a>

(5.19)

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M547">View MathML</a> in (5.19), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M548">View MathML</a>

which contradicts (5.14). The proof is complete. □

We finally remark that Theorem 1.1 is valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M549">View MathML</a>, from which one sees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/110/mathml/M62">View MathML</a> is a local minimizer of Φ.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The authors are grateful to the anonymous referee for his/her valuable suggestions. The second author was supported by NSFC11271264, NSFC11171204, KZ201010028027 and PHR201106118.

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