SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Gradient systems with sublinear term near the origin and asymptotically linear term near infinity

Hongrui Cai* and Jiabao Su

Author Affiliations

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

For all author emails, please log on.

Boundary Value Problems 2013, 2013:280  doi:10.1186/1687-2770-2013-280


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/280


Received:24 September 2013
Accepted:3 December 2013
Published:30 December 2013

© 2013 Cai 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 existence of nontrivial solutions for a sublinear gradient system with a nontrivial critical group at infinity.

MSC: 35J10, 35J65, 58E05.

Keywords:
gradient system; sublinear; critical group; Morse theory

1 Introduction

In this paper, we are concerned with the following gradient system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M1">View MathML</a>

(GS)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M2">View MathML</a> is a bounded open domain with a smooth boundary Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M3">View MathML</a> designates the partial derivative with respect to u of the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M4">View MathML</a>. The solutions of such systems are steady-states of reaction-diffusion systems arising in many applied sciences such as biology, chemistry, ecology or physics. It is well known that (GS) has variational structure when the nonlinearity F satisfies the subcritical growth condition

(F) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M5">View MathML</a> and there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M7">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M8">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M9">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M11">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M12">View MathML</a>.

That is, the solutions of (GS) can be found as critical points of the following functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M13">View MathML</a>

defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M14">View MathML</a> which is a Hilbert space endowed with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M15">View MathML</a>

and the associated norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M16">View MathML</a>

By the compact Sobolev embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M17">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M18">View MathML</a>, under the global assumption (F), the functional Φ is well defined and is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M19">View MathML</a> (see [1]) with its Fréchet derivative

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M20">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M21">View MathML</a>. The weak solutions to (GS) in E are exactly critical points of Φ in E.

We make some conventions. We use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M23">View MathML</a> to denote the norm and the inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M24">View MathML</a> and use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M25">View MathML</a> to denote an element in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M24">View MathML</a> and E. Bz denotes the matrix product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M27">View MathML</a> for a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M28">View MathML</a> matrix B and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M29">View MathML</a>. We use 0 to denote the origin in various spaces. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M30">View MathML</a> be the set of all continuous, cooperative and symmetric matrix functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M24">View MathML</a>. A matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M32">View MathML</a> takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M33">View MathML</a>

with the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M34">View MathML</a> satisfying the conditions that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M35">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M36">View MathML</a>, which means A is cooperative, and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M37">View MathML</a>.

When F satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M39">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M40">View MathML</a>, the system (GS) admits a trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M41">View MathML</a>. We are interested in the nontrivial solutions for (GS). In the current paper we apply the Morse theory to study the existence of nontrivial solutions of (GS) when the problem is sublinear near the origin and is asymptotically linear near infinity.

We make the following assumption near the origin.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M39">View MathML</a> and there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M46">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M47">View MathML</a>

In order to state the assumptions on the nonlinearity at infinity, we need some basic facts about the eigenvalue problem of linear gradient system. For a given matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M48">View MathML</a>, it is known (see [2,3]) that the corresponding linear system

admits a sequence of distinct eigenvalues of finite multiplicity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M50">View MathML</a>

such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M51">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M52">View MathML</a>. According to A, the space E can be split as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M53">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M54">View MathML</a>

The numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M56">View MathML</a> are well determined and finite.

We assume that the nonlinear system (GS) is asymptotically linear at infinity in the sense that the function F satisfies

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>) there is a matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M58">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M59">View MathML</a>

Associated to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M60">View MathML</a>, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M63">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M65">View MathML</a>. We say that the system (GS) is nonresonant at infinity if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M66">View MathML</a>, while it is resonant at infinity if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M67">View MathML</a>.

We first consider the nonresonance case. We have the following.

Theorem 1.1Assume thatFsatisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M70">View MathML</a>. Then (GS) has at least one nontrivial weak solution inE.

Next we consider the resonance case. We need additional assumptions on F near infinity.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M72">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M73">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M74">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M75">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M76">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M77">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M78">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M80">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M81">View MathML</a> imply that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M83">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M84">View MathML</a>

Theorem 1.2LetFsatisfy (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71">View MathML</a>). Then (GS) has at least one nontrivial weak solution inE. Moreover, ifFis even inz, then (GS) has infinitely many nontrivial weak solutions inE.

Theorem 1.3LetFsatisfy (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M74">View MathML</a>). Then (GS) has at least one nontrivial weak solution inE.

Now we give some remarks and comments. The gradient system represents the steady-state case of reaction-diffusion system which is a model for problems arising from biology, chemistry, physics and ecology, etc. In this paper we look for nontrivial solutions for the system (GS) via Morse theory. When the problem is resonant at infinity, we impose on the nonlinearity F the global assumption (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M91">View MathML</a>) to ensure the compactness and clear description of critical groups for Φ at infinity. (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M91">View MathML</a>) can be regarded as a variant of the famous Landesman-Lazer type resonance condition [4] which can be formulated as

See [5] for details. Near the origin we impose (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>), which means that ∇F is sublinear or F is sub-quadratic near zero. This kind of condition caught our attention first in a preprint by Liu and Wu [6] where a single elliptic equation was considered. This is the first use for gradient system in the current paper.

The asymptotically linear gradient systems (GS) have received some attention for years. We mention some recent related works [7-12] and the references therein. In these works, existence and multiplicity of nontrivial solutions for (GS) were obtained by combining various arguments involving Morse theory, saddle point reduction method (see [9-11]) and three critical point theorem (see [13]), etc. All above mentioned works dealt with the case that at least one of the critical groups of Φ at 0 is nontrivial somewhere. In the present paper, we study via Morse theory the case that all critical groups of Φ at 0 are trivial under the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>). Due to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>), the saddle point reduction methods [9-11] cannot be applied and there is no linking at 0. Comparing with known ones, the existence and multiplicity results for (GS) are all new. See more remarks in the last section of the paper.

The paper is organized as follows. In Section 2, we collect some basic abstract tools. In Section 3 we compute the critical groups at zero and infinity. The proofs of Theorems 1.1-1.3 and comments are given in Section 4.

2 Preliminary

In this section we cite some preliminaries that will be used to prove the main results of the paper. We first collect some results on Morse theory (see [14,15]) for a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M19">View MathML</a> functional Φ defined on a Hilbert space E.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M98">View MathML</a>. Denote for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M100">View MathML</a>

We say that Φ possesses the deformation property at the level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99">View MathML</a> if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M102">View MathML</a> and any neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M104">View MathML</a>, there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M105">View MathML</a> and a continuous deformation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M106">View MathML</a> such that

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M107">View MathML</a> for either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M108">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M109">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M110">View MathML</a> is non-increasing in t for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M111">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M112">View MathML</a>.

We say that Φ possesses the deformation property if Φ possesses the deformation property at each level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M113">View MathML</a>.

In applications the deformation property is ensured by the Palais-Smale condition or the Cerami condition.

We say that Φ satisfies the Palais-Smale condition at the level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99">View MathML</a> if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M115">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M117">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M118">View MathML</a> has a convergent subsequence. Φ satisfies the Palais-Smale condition if Φ satisfies the Palais-Smale condition at each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99">View MathML</a>. We say that Φ satisfies the Cerami condition [16,17] at the level <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99">View MathML</a> if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M115">View MathML</a> satisfying that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M123">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M124">View MathML</a> has a convergent subsequence. Φ satisfies the Cerami condition if Φ satisfies the Cerami condition at each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M99">View MathML</a>.

If Φ satisfies the Palais-Smale condition or the Cerami condition, then Φ possesses the deformation property [14,16].

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M126">View MathML</a> be an isolated critical point of Φ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M127">View MathML</a>, and U be a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M126">View MathML</a>. The group

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M129">View MathML</a>

is called the qth critical group of Φ at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M126">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M131">View MathML</a> denotes a singular relative homology group of the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M132">View MathML</a> with integer coefficients.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M133">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M134">View MathML</a> is bounded from below by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M135">View MathML</a> and Φ possesses the deformation property at all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M136">View MathML</a>. Then the group

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M137">View MathML</a>

is called the qth critical group of Φ at infinity [18].

Assume that Φ satisfies the deformation property and is a finite set. The Morse type numbers of the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M139">View MathML</a> are defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M140">View MathML</a>, and the Betti numbers of the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M139">View MathML</a> are defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M142">View MathML</a>.

Proposition 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M143">View MathML</a>possesses the deformation property, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M144">View MathML</a>, and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M146">View MathML</a>are finite and only finitely many of them are nonzero. Then it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M147">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M148">View MathML</a>

(2.2)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M149">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M150">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M151">View MathML</a>. From (2.1) one can deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M152">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M153">View MathML</a>. Thus if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M154">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M155">View MathML</a>, then Φ must have a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M156">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M157">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M158">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M159">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M160">View MathML</a>. Thus if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M161">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M160">View MathML</a>, then Φ must have a new critical point. Therefore the basic idea in applying Morse theory to find critical points of Φ is to compute critical groups both at infinity and at known critical points clearly and then to find unknown critical points by applying formulas (2.1) and (2.2).

Now we state an abstract result for the critical groups at infinity.

Proposition 2.2Let the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M163">View MathML</a>take the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M164">View MathML</a>

(2.3)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M165">View MathML</a>is a self-adjoint linear operator such that 0 is isolated in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M166">View MathML</a>, the spectrum of ℒ. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M167">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M168">View MathML</a>

(2.4)

Denote<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M170">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M171">View MathML</a>are subspaces on whichis positive (negative) definite. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M172">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M173">View MathML</a>are finite, and Φ possesses the deformation property.

(1) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M174">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M175">View MathML</a>

(2) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M176">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M177">View MathML</a>

provided Φ satisfies the angle conditions with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M178">View MathML</a>:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M179">View MathML</a>) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M180">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M181">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M182">View MathML</a>

Proposition 2.1(1) was obtained in [19] (see Remark 5.2 in [14]). Proposition 2.1(2) is a revision of Proposition 3.10 in [18] which was made first in [20] and was remade in [21].

Next we recall an abstract critical point theorem built by Wang in [22].

Proposition 2.3 ([22])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M183">View MathML</a>, whereXis a Banach space. Assume that Φ possesses the deformation property, is even and bounded from below, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M184">View MathML</a>. If for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M185">View MathML</a>, there existk-dimensional subspaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M186">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M187">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M188">View MathML</a>

(2.5)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M189">View MathML</a>, then Φ has a sequence of critical values<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M190">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M191">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M52">View MathML</a>.

Finally, we mention the eigenvalues of the linear gradient system (LA). By the compact embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M193">View MathML</a>, for a given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M194">View MathML</a>, there is a compact self-adjoint operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M195">View MathML</a> associated with A such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M196">View MathML</a>

The operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M197">View MathML</a> possesses the property that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M198">View MathML</a> is an eigenvalue of (LA) if and only if there is nonzero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M199">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M200">View MathML</a>

(LA) has a sequence of distinct eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M201">View MathML</a>

and each eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M198">View MathML</a> of (LA) has a finite multiplicity. All eigenvectors of (LA) form a Hilbertian basis of E and that E can be split as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M203">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M205">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M206">View MathML</a> are the negative, positive definite invariant subspaces and the kernel of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M207">View MathML</a>, respectively. We refer to [2,3] for more properties related to the eigenvalue problem (LA) and the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M197">View MathML</a>.

3 Critical groups and compactness

In this section we verify the compactness of the functional Φ and compute the critical groups of Φ at both zero and infinity. Without loss of generality, we assume that (GS) has finitely many weak solutions so that the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M41">View MathML</a> is an isolated critical point of Φ. We first compute the critical groups <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M210">View MathML</a>. The idea was from an unpublished preprint by Liu and Wu [6] where a single elliptic equation was studied.

We work with the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M211">View MathML</a>

Lemma 3.1Assume thatFsatisfies (F) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M213">View MathML</a>

(3.1)

Proof Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M214">View MathML</a>. By definition of critical groups, we can write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M215">View MathML</a>. We will construct a deformation mapping from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M216">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M217">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M218">View MathML</a> small.

In the following we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M219">View MathML</a> to denote positive constants. By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>), one deduces that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M221">View MathML</a>

(3.2)

It follows from (F) and (2.2) that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M222">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M223">View MathML</a>

(3.3)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M224">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M225">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M226">View MathML</a>

(3.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M227">View MathML</a>, for each given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M228">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M229">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M230">View MathML</a>

(3.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M231">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M232">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M233">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M234">View MathML</a>

(3.6)

From (3.6), one concludes that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M235">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M236">View MathML</a>

(3.7)

From now on we fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M235">View MathML</a>. We claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M238">View MathML</a>

(3.8)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M240">View MathML</a>. By the continuity of Φ, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M241">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M242">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M243">View MathML</a>. We will get (3.8) by proving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M244">View MathML</a>. Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M245">View MathML</a>. Then there is some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M246">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M247">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M248">View MathML</a>, it follows from (3.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M249">View MathML</a>

(3.9)

But <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M250">View MathML</a> implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M251">View MathML</a>

(3.10)

This contradicts (3.9). Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M244">View MathML</a> and (3.8) holds.

Now define a mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M253">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M254">View MathML</a>

(3.11)

By (3.5), (3.7) and (3.8), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M255">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M256">View MathML</a>, there exists a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M257">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M258">View MathML</a>

(3.12)

Thus the mapping π is well defined. Moreover, it follows from (3.7), (3.12) and the implicit function theorem that the mapping π is continuous in z. Define a mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M259">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M260">View MathML</a>

Then η is a continuous deformation from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M261">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M262">View MathML</a>. By homotopy invariance of a homology group and the contractibility of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M263">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M264">View MathML</a>

The proof is complete. □

We remark here that in [23] the similar idea for computing the critical groups at 0 was presented for a single elliptic equation. For (GS), the conditions used in [23] can be formulated as

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M265">View MathML</a>) there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M46">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M268">View MathML</a>

We note here that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M265">View MathML</a>) is not comparable with (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>) since (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>) is a local condition and although (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>) implies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M265">View MathML</a>)(i) but (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M265">View MathML</a>)(ii) is a global condition.

Now we verify the compactness for the functional Φ and compute the critical groups of Φ at infinity. To do this, we rewrite the functional Φ as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M275">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M276">View MathML</a>.

Lemma 3.2LetFsatisfy (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71">View MathML</a>).

(i) The functional Φ is coercive onEand satisfies the Palais-Smale condition.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M280">View MathML</a>.

Proof (i) First, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>) implies (F) while (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71">View MathML</a>) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M283">View MathML</a>

(3.13)

We will prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M284">View MathML</a>

(3.14)

Assume that there is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M285">View MathML</a> such that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M180">View MathML</a>, it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M287">View MathML</a>

(3.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M288">View MathML</a>

(3.16)

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M289">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M290">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M291">View MathML</a>. Up to a subsequence, we may assume that there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M292">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M293">View MathML</a>

(3.17)

By (3.13) one has that for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M294">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M295">View MathML</a>

Therefore by (3.14) we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M296">View MathML</a>

(3.18)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M118">View MathML</a> in (3.18), it follows from (3.15) and (3.16) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M298">View MathML</a>

(3.19)

On the other hand, we have by the lower semi-continuity of the norm that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M299">View MathML</a>

(3.20)

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M300">View MathML</a>

(3.21)

By (3.16), (3.21) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M301">View MathML</a>

(3.22)

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M302">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M303">View MathML</a> is an eigenvector corresponding to the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M72">View MathML</a>. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M305">View MathML</a> for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M306">View MathML</a> and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M307">View MathML</a>

(3.23)

Now it follows from (3.13), (3.23) and the Fatou lemma that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M308">View MathML</a>

(3.24)

This is a contradiction. Thus Φ is coercive on E.

By the coercivity of Φ, a Palais-Smale sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M309">View MathML</a> of Φ must be bounded. Since F has a subcritical growth, a standard argument shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M310">View MathML</a> has a convergent subsequence.

(ii) Since Φ is coercive and weakly lower semi-continuous, Φ is bounded from below. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M311">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M312">View MathML</a>

The proof is complete. □

Lemma 3.3LetFsatisfy (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M74">View MathML</a>).

(i) Φ satisfies the Cerami condition.

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M315">View MathML</a>if (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M316">View MathML</a>) holds.

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M317">View MathML</a>if (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M318">View MathML</a>) holds.

Proof (i) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M319">View MathML</a> be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M320">View MathML</a>

(3.25)

We only need to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M309">View MathML</a> is bounded in E. Suppose, by the way of contradiction, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M322">View MathML</a>

(3.26)

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M323">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M324">View MathML</a>. Passing to a subsequence if necessary, we may assume that there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M325">View MathML</a> such that as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M118">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M327">View MathML</a>

(3.27)

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>) and (3.26) we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M329">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M330">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M331">View MathML</a>

Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M332">View MathML</a>

(3.28)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M333">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M334">View MathML</a>

(3.29)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M335">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M336">View MathML</a>

(3.30)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M337">View MathML</a> in (3.29) and using (3.25), (3.28), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M338">View MathML</a>

(3.31)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M339">View MathML</a> in (3.30), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M340">View MathML</a>

(3.32)

Therefore <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M341">View MathML</a> and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M342">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M343">View MathML</a> is an eigenvector of (L<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M344">View MathML</a>) associated with eigenvalue 1. It follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M345">View MathML</a>. Write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M346">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M347">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M348">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M349">View MathML</a>

(3.33)

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M74">View MathML</a>) there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M351">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M352">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M353">View MathML</a>

This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M354">View MathML</a>

This is a contradiction with (3.25).

(ii) We apply Proposition 2.2. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M355">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M356">View MathML</a>

Then Φ can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M357">View MathML</a>

(3.34)

By Lemma 3.1, Φ satisfies the Cerami condition and hence possesses the deformation property. From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>) one sees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M359">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M360">View MathML</a>

Now we show that Φ satisfies the angle condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M361">View MathML</a>) at infinity with respect to the orthogonal decomposition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M362">View MathML</a> when (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M316">View MathML</a>) holds. Suppose it is not true, then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M364">View MathML</a>, there are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M365">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M366">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M347">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M368">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M369">View MathML</a>

(3.35)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M370">View MathML</a>

(3.36)

It follows from (3.35) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M371">View MathML</a>

By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M316">View MathML</a>) we have that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M373">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M374">View MathML</a>

this contradicts (3.36). Therefore (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M361">View MathML</a>) holds, and by Proposition 2.2 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M376">View MathML</a>

(iii) This case is proved in a similar way.

The proof is finished. □

4 Proofs of main theorems

In this section we give the proofs of main theorems in this paper.

Proof of Theorem 1.1 By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>), the functional Φ takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M378">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M379">View MathML</a> is a bounded self-adjoint linear operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M167">View MathML</a> with a compact gradient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M359">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M382">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M70">View MathML</a>, the problem is no resonance at infinity and thus Ψ satisfies the Palais-Smale condition. By Proposition 2.2(1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M384">View MathML</a>

(4.1)

Therefore Φ has a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M385">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M386">View MathML</a>

(4.2)

By Lemma 3.1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M387">View MathML</a>

(4.3)

By (4.2) and (4.3), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M388">View MathML</a> and then is a nontrivial weak solution of (GS). □

Proof of Theorem 1.2 By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71">View MathML</a>) and Lemma 3.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M391">View MathML</a>

(4.4)

Therefore Φ has a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M385">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M393">View MathML</a>

(4.5)

We still have (4.3). Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M388">View MathML</a> is a nontrivial weak solution of (GS). In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M156">View MathML</a> is a global minimizer of Φ.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M396">View MathML</a> is even in z. We will employ Proposition 2.3 to prove the multiplicity in Theorem 1.2. Now Φ is even, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M184">View MathML</a>. By Lemma 3.2, Φ satisfies the Palais-Smale condition and is bounded from below following from the coercivity.

We verify (2.5). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M398">View MathML</a> be a k-dimensional subspace of E. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M399">View MathML</a>, as arguments in the proof of Lemma 3.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M400">View MathML</a>

(4.6)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M401">View MathML</a> and all norms on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M398">View MathML</a> are equivalent, we get that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M187">View MathML</a> small enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M404">View MathML</a>

With all the conditions of Proposition 2.3 being verified, we get the conclusion that Φ has a sequence of critical values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M190">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M406">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M52">View MathML</a>. Thus (GS) has infinitely many nontrivial weak solutions in E. The proof is finished. □

Proof of Theorem 1.3 By a similar argument, it follows from Lemma 3.1 and Lemma 3.3. □

We conclude the paper with further comments and remarks.

Remark 4.1 (i) In Theorem 1.1, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M408">View MathML</a> which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M409">View MathML</a> and F is even in z, by the same arguments as the last part of the proof of Theorem 1.2, one can show that (GS) has infinitely many nontrivial weak solutions in E with negative energies which converge to zero.

(ii) In Theorem 1.2, one nontrivial solution could be obtained if (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71">View MathML</a>) is replaced by the nonquadraticity condition [24]

Indeed, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M412">View MathML</a>) is equivalent to

which implies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71">View MathML</a>), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M415">View MathML</a>

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M71">View MathML</a>) is weaker than (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M417">View MathML</a>).

(iii) The result for one nontrivial solution in Theorem 1.2 is valid when F satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M42">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M57">View MathML</a>) and the nonquadraticity condition [24]

Indeed, in this case, Φ satisfies the Cerami condition and Φ has a saddle point structure at infinity with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M362">View MathML</a> in the sense that Φ is bounded from below on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M422">View MathML</a> and is anti-coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M423">View MathML</a>. Then Proposition 3.8 in [18] is applied to get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M424">View MathML</a>.

(iv) In Theorem 1.3, the global condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M425">View MathML</a>) is somewhat abstract and has been used in [12]. It could be verified if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M426">View MathML</a> acts as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M427">View MathML</a> near infinity for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M428">View MathML</a> (see [20,25]). See [12] for more comparisons.

Remark 4.2 In Theorem 1.2, we proved the multiplicity result by a critical point theorem in [22] when Φ is even. This result is completely new for gradient systems. Since the critical groups of Φ at both zero and infinity are clearly computed, when Φ is even, the Morse equality may provide us an idea to give a different proof provided we have in hand the following basic conclusion.

(⋆) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M156">View MathML</a> is a solution of (HS), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M430">View MathML</a> for finitely many <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M431">View MathML</a>.

Let (⋆) hold and let F be even. We prove the multiplicity for (GS) in Theorem 1.2 via Morse theory. Assume that (GS) has only finitely many pairs of nontrivial solutions. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M432">View MathML</a>. Then by the Morse equality, one has that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M433">View MathML</a>

By (⋆), (4.3) and (4.4), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M434">View MathML</a>

a contradiction. Similarly, if (⋆) is valid and F is even, then we have the same multiplicity result in Theorems 1.1 and 1.3.

We note here that the conclusion (⋆) is valid for Φ is of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M435">View MathML</a>. A natural problem arises here whether or not that (⋆) is valid for a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/280/mathml/M19">View MathML</a> functional. It is still open to the best of our knowledge. We will focus on this problem in near future.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to the manuscript and read and approved the final manuscript.

Acknowledgements

Supported by NSFC11271264, NSFC11171204 and PHR201106118.

References

  1. Rabinowitz, P: Minimax Methods in Critical Point Theory with Application to Differential Equations, Am. Math. Soc., Providence (1986)

  2. Chang, KC: An extension of the Hess-Kato theorem to elliptic systems and its applications to multiple solutions problems. Acta Math. Sin.. 15, 439–454 (1999). Publisher Full Text OpenURL

  3. Chang, KC: Principal eigenvalue for weight in elliptic systems. Nonlinear Anal.. 46, 419–433 (2001). Publisher Full Text OpenURL

  4. Landesman, E, Lazer, A: Nonlinear perturbations of linear eigenvalues problem at resonance. J. Math. Mech.. 19, 609–623 (1970)

  5. Bartsch, T, Chang, KC, Wang, Z-Q: On the Morse indices of sign changing solution of nonlinear elliptic problem. Math. Z.. 233, 655–677 (2000). Publisher Full Text OpenURL

  6. Liu, JQ, Wu, SP: A note on a class of sublinear elliptic equation. Research Report 84, Peking University (1997)

  7. da Silva, ED: Multiplicity of solutions for gradient systems. Electron. J. Differ. Equ.. 2010, Article ID 64 (2010)

  8. da Silva, ED: Multiplicity of solutions for gradient systems using Landesman-Lazer conditions. Abstr. Appl. Anal.. 2010, Article ID 237826 (2010)

  9. Furtado, MF, de Paiva, FOV: Multiplicity of solutions for resonant elliptic systems. J. Math. Anal. Appl.. 319, 435–449 (2006). Publisher Full Text OpenURL

  10. Furtado, MF, de Paiva, FOV: Multiple solutions for resonant elliptic systems via reduction method. Bull. Aust. Math. Soc.. 82, 211–220 (2010). Publisher Full Text OpenURL

  11. Li, C, Liu, S: Homology of saddle point reduction and applications to resonant elliptic systems. Nonlinear Anal.. 81, 236–246 (2013)

  12. Lü, L, Su, J: Solutions to a gradient system with resonance at both zero and infinity. Nonlinear Anal.. 74, 5340–5351 (2011). Publisher Full Text OpenURL

  13. Liu, JQ, Su, J: Remarks on multiple nontrivial solutions for quasi-linear resonant problems. J. Math. Anal. Appl.. 258, 209–222 (2001). Publisher Full Text OpenURL

  14. Chang, KC: Infinite Dimensional Morse Theory and Multiple Solutions Problems, Birkhäuser, Boston (1993)

  15. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems, Springer, Berlin (1989)

  16. Bartolo, P, Benci, V, Fortunato, D: Abstract critical point theorems and applications to nonlinear problems with ‘strong’ resonance at infinity. Nonlinear Anal.. 7, 981–1012 (1983). Publisher Full Text OpenURL

  17. Cerami, G: Un criterio di esistenza per i punti critici su varietà illimitate. Rend. - Ist. Lomb., Accad. Sci. Lett., a Sci. Mat. Fis. Chim. Geol.. 112, 332–336 (1978)

  18. Bartsch, T, Li, S: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal.. 28, 419–441 (1997). Publisher Full Text OpenURL

  19. Wang, Z-Q: Multiple solutions for indefinite functionals and applications to asymptotically linear problems. Acta Math. Sin. New Ser.. 5(2), 101–113 (1989). Publisher Full Text OpenURL

  20. Su, J: Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity. J. Differ. Equ.. 145(2), 252–273 (1998). Publisher Full Text OpenURL

  21. Su, J, Zhao, L: An elliptic resonance problem with multiple solutions. J. Math. Anal. Appl.. 319, 604–616 (2006). Publisher Full Text OpenURL

  22. Wang, Z-Q: Nonlinear boundary value problems with concave nonlinearities near the origin. Nonlinear Differ. Equ. Appl.. 8, 15–33 (2001). Publisher Full Text OpenURL

  23. Moroz, V: Solutions of superlinear at zero elliptic equations via Morse theory. Topol. Methods Nonlinear Anal.. 10, 387–397 (1997)

  24. Costa, DG, Magalhães, CA: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal.. 23, 1401–1412 (1994). Publisher Full Text OpenURL

  25. Su, J, Tang, C: Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues. Nonlinear Anal.. 44, 311–321 (2001). Publisher Full Text OpenURL