SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Existence of solutions for a class of biharmonic equations with the Navier boundary value condition

Ruichang Pei

Author Affiliations

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

Boundary Value Problems 2012, 2012:154  doi:10.1186/1687-2770-2012-154


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/154


Received:18 July 2012
Accepted:14 December 2012
Published:28 December 2012

© 2012 Pei; 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, the existence of at least one nontrivial solution for a class of fourth-order elliptic equations with the Navier boundary value conditions is established by using the linking methods.

Keywords:
biharmonic; Navier boundary value problems; local linking

1 Introduction

Consider the following Navier boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M2">View MathML</a> is the biharmonic operator, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M3">View MathML</a> and Ω is a bounded smooth domain in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M5">View MathML</a>).

The conditions imposed on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M6">View MathML</a> are as follows:

(H1) , and there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M8">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M9">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M10">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M12">View MathML</a>, uniformly on Ω;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M13">View MathML</a> uniformly on Ω;

(H4) There is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M14">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M16">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M17">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M18">View MathML</a>;

(H5) For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M19">View MathML</a>, either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M20">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M21">View MathML</a>

This fourth-order semilinear elliptic problem has been studied by many authors. In [1], there was a survey of results obtained in this direction. In [2], Micheletti and Pistoia showed that (1.1) admits at least two solutions by a variation of linking if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M22">View MathML</a> is sublinear. And in [3], the authors proved that the problem (1.1) has at least three solutions by a variational reduction method and a degree argument. In [4], Zhang and Li showed that (1.1) admits at least two nontrivial solutions by the Morse theory and local linking if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M22">View MathML</a> is superlinear and subcritical on u. In [5], Zhang and Wei obtained the existence of infinitely many solutions for the problem (1.1) where the nonlinearity involves a combination of superlinear and asymptotically linear terms. As far as the problem (1.1) is concerned, existence results of sign-changing solutions were also obtained. See, e.g., [6,7] and the references therein.

We will use linking methods to give the existence of at least one nontrivial solution for (1.1).

Let X be a Banach space with a direct sum decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M24">View MathML</a>

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M25">View MathML</a> has a local linking at 0, with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M26">View MathML</a> if for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M27">View MathML</a>,

(1.2)

(1.3)

It is clear that 0 is a critical point of I.

The notion of local linking generalizes the notions of local minimum and local maximum. When 0 is a non-degenerate critical point of a functional of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M30">View MathML</a> defined on a Hilbert space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M31">View MathML</a>, I has local linking at 0.

The condition of local linking was introduced in [8] under stronger assumptions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M32">View MathML</a>

Under those assumptions, the existence of nontrivial critical points was proved for functionals which are

(a) bounded below [8],

(b) superquadratic [8] and

(c) asymptotically quadratic [9].

The cases (a), (b) and (c) were considered in [10] with only conditions (1.2) and (1.3), and three theorems about critical points were proved. Using these theorems, the author in [10] proved the existence of at least one nontrivial solution for the second-order elliptic boundary value problem with the Dirichlet boundary value condition. In the present paper, we also use the three theorems in [10] and the linking technique to give the existence of at least one nontrivial solution for the biharmonic problem (1.1) under a weaker condition. The main results are as follows.

Theorem 1.1Assume the conditions (H1)-(H4) hold. Iflis an eigenvalue of −△ (with the Dirichlet boundary condition), assume also (H5). Then the problem (1.1) has at least one nontrivial solution.

We also consider asymptotically quadratic functions.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M33">View MathML</a> be the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M34">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M35">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M36">View MathML</a>) is the eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M37">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M38">View MathML</a>. We assume that

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M40">View MathML</a>, uniformly in Ω, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M41">View MathML</a>.

Theorem 1.2Assume the conditions (H1), (H6) and one of the following conditions:

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M43">View MathML</a>;

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M44">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M43">View MathML</a>, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M19">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M47">View MathML</a>

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M48">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M43">View MathML</a>, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M19">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M51">View MathML</a>

Then the problem (1.1) has at least one nontrivial solution.

2 Preliminaries

Let X be a Banach space with a direct sum decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M52">View MathML</a>

Consider two sequences of a subspace:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M53">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M54">View MathML</a>

For every multi-index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M55">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M56">View MathML</a>. We know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M57">View MathML</a>

A sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M58">View MathML</a> is admissible if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M59">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M60">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M61">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M62">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M63">View MathML</a> the function I restricted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M64">View MathML</a>.

Definition 2.1 Let I be locally Lipschitz on X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M65">View MathML</a>. The functional I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M66">View MathML</a> condition if every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M67">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M68">View MathML</a> is admissible and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M69">View MathML</a>

contains a subsequence which converges to a critical point of I.

Definition 2.2 Let I be locally Lipschitz on X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M65">View MathML</a>. The functional I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M71">View MathML</a> condition if every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M67">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M68">View MathML</a> is admissible and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M74">View MathML</a>

contains a subsequence which converges to a critical point of I.

Remark 2.1

1. The <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M71">View MathML</a> condition implies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M66">View MathML</a> condition for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M65">View MathML</a>.

2. When the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M66">View MathML</a> sequence is bounded, then the sequence is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M79">View MathML</a> sequence (see [11]).

3. Without loss of generality, we assume that the norm in X satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M80">View MathML</a>

Definition 2.3 Let X be a Banach space with a direct sum decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M81">View MathML</a>

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M25">View MathML</a> has a local linking at 0, with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M83">View MathML</a>, if for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M27">View MathML</a>,

Lemma 2.1 (see [10])

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M25">View MathML</a>satisfies the following assumptions:

(B1) Ihas a local linking at 0 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M87">View MathML</a>;

(B2) Isatisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M88">View MathML</a>;

(B3) Imaps bounded sets into bounded sets;

(B4) for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M92">View MathML</a>. ThenIhas at least two critical points.

Remark 2.2 Assume I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M66">View MathML</a> condition. Then this theorem still holds.

Let X be a real Hilbert space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M25">View MathML</a>. The gradient of I has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M95">View MathML</a>

where A is a bounded self-adjoint operator, 0 is not the essential spectrum of A, and B is a nonlinear compact mapping.

We assume that there exist an orthogonal decomposition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M96">View MathML</a>

and two sequences of finite-dimensional subspaces,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M97">View MathML</a>

such that

For every multi-index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M55">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M64">View MathML</a> the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M101">View MathML</a>

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M102">View MathML</a> the orthogonal projector onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M64">View MathML</a>, and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M104">View MathML</a> the Morse index of a self-adjoint operator L.

Lemma 2.2 (see [10])

Isatisfies the following assumptions:

(i) Ihas a local linking at 0 with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M83">View MathML</a>;

(ii) there exists a compact self-adjoint operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M106">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M107">View MathML</a>

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M108">View MathML</a>is invertible;

(iv) for infinitely many multiple-indices<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M109">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M110">View MathML</a>

ThenIhas at least two critical points.

3 The proof of main results

Proof of Theorem 1.1 (1) We shall apply Lemma 2.1 to the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M111">View MathML</a>

defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M112">View MathML</a>. We consider only the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M113">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M114">View MathML</a>

(3.1)

Then other case is similar and simple.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M115">View MathML</a> be the finite dimensional space spanned by the eigenfunctions corresponding to negative eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M116">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M117">View MathML</a> be its orthogonal complement in X. Choose a Hilbertian basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M118">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M119">View MathML</a>) for X and define

By the condition (H1) and Sobolev inequalities, it is easy to see that the functional I belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M121">View MathML</a> and maps bounded sets to bounded sets.

(2) We claim that I has a local linking at 0 with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M83">View MathML</a>. Decompose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M117">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M124">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M126">View MathML</a>. Also, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M130">View MathML</a>. By the equivalence of norm in the finite-dimensional space, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M131">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M132">View MathML</a>

(3.2)

It follows from (H1) and (H2) that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M133">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M134">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M135">View MathML</a>

(3.3)

Hence, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M136">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M138">View MathML</a> is a constant and hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M27">View MathML</a> small enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M140">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M141">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M142">View MathML</a> and let

From (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M144">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M145">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M146">View MathML</a>. On the one hand, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M147">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M148">View MathML</a>. Hence, from (H5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M149">View MathML</a>

On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M150">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M151">View MathML</a>. It follows from (3.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M152">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M153">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M154">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M145">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M156">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M157">View MathML</a> is a constant. Hence, there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M159">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M160">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M161">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M128">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M145">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M164">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M165">View MathML</a> small enough.

(3) We claim that I satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M66">View MathML</a>. Consider a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M67">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M67">View MathML</a> is admissible and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M169">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M170">View MathML</a>

(3.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M171">View MathML</a>. Up to a subsequence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M172">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M173">View MathML</a>, we choose a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M174">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M175">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M137">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M177">View MathML</a>. By the Sobolev imbedded theory, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M178">View MathML</a>

So, for n large enough, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M179">View MathML</a>, and combining Ehrling-Nirenberg-Gagliardo inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M180">View MathML</a>

(3.6)

where ϵ is a small enough constant.

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M181">View MathML</a>. Now, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M183">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M184">View MathML</a> and

(3.7)

Therefore, using (H3), we have

This contradicts (3.5).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M187">View MathML</a>, then the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M188">View MathML</a> has a positive Lebesgue measure. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M189">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M190">View MathML</a>. Hence, by (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M191">View MathML</a>

(3.8)

From (3.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M192">View MathML</a>

(3.9)

By (3.8), the right-hand side of (3.9) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M193">View MathML</a>. This is a contradiction.

In any case, we obtain a contradiction. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M194">View MathML</a> is bounded.

Finally, we claim that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M89">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M196">View MathML</a>

By (H2) and (H3), there exists large enough M such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M197">View MathML</a>

So, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M198">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/154/mathml/M199">View MathML</a>

Hence, our claim holds. □

Proof of Theorem 1.2 We omit the proof which depends on Lemma 2.2 and is similar to the preceding one. □

Competing interests

The author declares that he has no competing interests.

Author’s contributions

The author read and approved the final manuscript.

Acknowledgements

The author would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant No. 10671156) of China.

References

  1. Lazer, AC, Mckenna, PJ: Large amplitude periodic oscillation in suspension bridges: some new connections with nonlinear analysis. SIAM Rev.. 32, 537–578 (1990). Publisher Full Text OpenURL

  2. Micheletti, AM, Pistoia, A: Multiplicity solutions for a fourth order semilinear elliptic problems. Nonlinear Anal. TMA. 31, 895–908 (1998). Publisher Full Text OpenURL

  3. Chipot, M: Variational Inequalities and Flow in Porous Media, Springer, New York (1984)

  4. Zhang, JH, Li, SJ: Multiple nontrivial solutions for some fourth-order semilinear elliptic problems. Nonlinear Anal. TMA. 60, 221–230 (2005)

  5. Zhang, J, Wei, ZL: Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems. Nonlinear Anal. TMA. 74, 7474–7485 (2011). Publisher Full Text OpenURL

  6. Zhou, JW, Wu, X: Sign-changing solutions for some fourth-order nonlinear elliptic problems. J. Math. Anal. Appl.. 342, 542–558 (2008). Publisher Full Text OpenURL

  7. Liu, XQ, Huang, YS: One sign-changing solution for a fourth-order asymptotically linear elliptic problem. Nonlinear Anal. TMA. 72, 2271–2276 (2010). Publisher Full Text OpenURL

  8. Liu, JQ, Li, SJ: Some existence theorems on multiple critical points and their applications. Chin. Sci. Bull.. 17, 1025–1027 (1984)

  9. Li, SJ, Liu, JQ: Morse theory and asymptotic linear Hamiltonian systems. J. Differ. Equ.. 78, 53–73 (1989). Publisher Full Text OpenURL

  10. Li, SJ, Willem, M: Applications of local linking to critical point theory. J. Math. Anal. Appl.. 189, 6–32 (1995). Publisher Full Text OpenURL

  11. Teng, KM: Existence and multiplicity results for some elliptic systems with discontinuous nonlinearities. Nonlinear Anal. TMA. 75, 2975–2987 (2012). Publisher Full Text OpenURL