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Open Access Research

Multiple solutions for a fourth-order nonlinear elliptic problem

Ruichang Pei

Author Affiliations

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

Boundary Value Problems 2012, 2012:39  doi:10.1186/1687-2770-2012-39

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/39


Received:24 December 2011
Accepted:5 April 2012
Published:5 April 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

The existence of multiple solutions for a class of fourth-order elliptic equation with respect to the generalized asymptotically linear conditions is established by using the minimax method and Morse theory.

Keywords:
fourth-order elliptic boundary value problems; multiple solutions; mountain pass theorem; Morse theory.

1 Introduction

Consider the following Navier boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M1">View MathML</a>

(1)

where Δ2 is the biharmonic operator, and Ω is a bounded smooth domain in ℝN (N > 4), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M2">View MathML</a> the first eigenvalue of -Δ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M3">View MathML</a>.

The conditions imposed on f (x, t) are as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M4">View MathML</a> for all x ∈ Ω, t ∈ ℝ;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a> f1(t) ≤ f (x, t) ≤ f2 (t) uniformly in x ∈ Ω, where fl, f2 C (ℝ) and we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M6">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M7">View MathML</a>

In view of the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a>, problem (1) is called generalized asymptotically linear at both zero and infinity. Clearly, u = 0 is a trivial solution of problem (1). It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a> that the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M9">View MathML</a>

(2)

is of C2 on the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M10">View MathML</a> with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M11">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M12">View MathML</a> Under the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a>, the critical points of I are solutions of problem (1). Let 0 < λ1 < λ2 < ··· < λk < ··· be the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M13">View MathML</a> and ϕl(x) > 0 be the eigenfunction corresponding to λl. In fact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M14">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M15">View MathML</a> denote the eigenspace associated to λk. Throughout this article, we denoted by | · |p the Lp (Ω) norm.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M16">View MathML</a> in the above condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a> is an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M17">View MathML</a>, then the problem (1) is called resonance at infinity. Otherwise, we call it non-resonance. A main tool of seeking the critical points of functional I is the mountain pass theorem (see [1-3]). To apply this theorem to the functional I in (2), usually we need the following condition [1], i.e., for some θ > 2 and M > 0,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M18">View MathML</a>

It is well known that the condition (AR) plays an important role in verifying that the functional I has a "mountain-pass" geometry and a related (PS)c sequence is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19">View MathML</a> when one uses the mountain pass theorem.

If f(x, t) admits subcritical growth and satisfies (AR) condition by the standard argument of applying mountain pass theorem, we know that problem (1) has nontrivial solutions. Similarly, lase f(x, t) is of critical growth (see, for example, [4-7] and their references).

It follows from the condition (AR) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M20">View MathML</a> after a simple computation. That is, f(x, t) must be superlinear with respect to t at infinity. Noticing our condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a>, the nonlinear term f(x, t) is generalized asymptotically linear, not superlinear, with respect to t at infinity; which means that the usual condition (AR) cannot be assumed in our case. If the mountain pass theorem is used to seek the critical points of I, it is difficult to verify that the functional I has a "mountain pass" structure and the (PS)c sequence is bounded.

In [8], Zhou studied the following elliptic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M21">View MathML</a>

where the conditions on f(x, t) are similar to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a>. He provided a valid method to verify the (PS) sequence of the variational functional, for above problem is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M22">View MathML</a> (see also [9,10]).

To the authors' knowledge, there seems few results on problem (1) when f(x, t) is generalized asymptotically linear at infinity. However, the method in [8] cannot be applied directly to the biharmonic problems. For example, for the Laplacian problem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M23">View MathML</a> implies |u|, u+, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M24">View MathML</a>, where u+ = max(u, 0), u- = max(-u, 0). We can use u+ or u- as a test function, which is helpful in proving a solution nonnegative. While for the biharmonic problems, this trick fails completely since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M25">View MathML</a> does not imply u+, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M26">View MathML</a> (see [[11], Remark 2.1.10] and [12,13]). As far as this point is concerned, we will make use of the new methods to discuss in the following Lemma 2.2.

This fourth-order semilinear elliptic problem can be considered as an analogue of a class of second-order problems which have been studied by many authors. In [14], there was a survey of results obtained in this direction. In [15], Micheletti and Pistoia showed that (P1) admits at least two solutions by a variation of linking if f(x, u) is sublinear. Chipot [16] proved that the problem (P1) has at least three solutions by a variational reduction method and a degree argument. In [17], Zhang and Li showed that (P1) admits at least two nontrivial solutions by Morse theory and local linking if f(x, u) is superlinear and subcritical on u.

In this article, we consider multiple solutions of problem (1) in the non-resonance by using the mountain pass theorem and Morse theory. At first, we use the truncated skill and mountain pass theorem to obtain a positive solution and a negative solution of problem (1) under our more general conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a> with respect to the conditions (H1) and (H3) in [8]. In the course of proving existence of positive solution and negative solution, our conditions are general, but the proof of our compact condition is more simple than that in [8]. Furthermore, we can obtain a nontrivial solution when the nonlinear term f is non-resonance at the infinity by using Morse theory.

2 Main result and auxiliary lemmas

Let us now state the main result.

Theorem 2.1. Assume conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a>hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M27">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M28">View MathML</a>for some k ≥ 2, then problem (1) has at least three nontrivial solutions.

Consider the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M29">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M30">View MathML</a>

Define a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M31">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M32">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M33">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M34">View MathML</a>.

Lemma 2.2. I+ satisfies the (PS) condition.

Proof. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M35">View MathML</a> be a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M36">View MathML</a> as n → ∞. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M37">View MathML</a>

(3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M38">View MathML</a>. Assume that |un|2 is bounded, taking ϕ = un in (3). By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a>, there exists c > 0 such that |f+(x, un(x))| ≤ c|un(x)|, a.e. x ∈ Ω. So (un) is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19">View MathML</a>. If |un|2 → +∞, as n → ∞, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M39">View MathML</a>, then |vn|2 = 1. Taking ϕ = vn in (3), it follows that ||vn|| is bounded. Without loss of generality, we assume that vn v in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19">View MathML</a>, then vn→ v in L2(Ω). Hence, vn → v a.e. in Ω. Dividing both sides of (3) by |un|2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M40">View MathML</a>

(4)

Let ϵ > 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M41">View MathML</a>, and choose a constant M > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M42">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M38">View MathML</a> and ϕ ≥ 0. From (4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M43">View MathML</a>

Letting n → ∞, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M38">View MathML</a> and ϕ ≥ 0 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M44">View MathML</a>

(5)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M45">View MathML</a>

While let -Δv = u, by the comparison maximum principle v ≥ 0. Since the definition of vn, we have v ≢ 0 and we arrive at a contradiction by choosing ϕ = ϕ1 in (5).

Since |un|2 is bounded, from (3) we get the boundedness of ||un||. A standard argument shows that {un} has a convergent subsequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19">View MathML</a>. Therefore, I+ satisfies (PS) condition.

Lemma 2.3. Let ϕ1 be the eigenfunction corresponding to λ1 with ||ϕ1|| = 1. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M46">View MathML</a>, then

(a) There exist ρ, β > 0 such that I+ (u) ≥ β for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M47">View MathML</a>with ||u|| = ρ;

(b) I+(1) = -∞ as t → +∞.

Proof. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M48">View MathML</a>, for any ε > 0, there exist A = A(ε) ≥ 0 and B = B(ε) such that for all (x, s) ∈ Ω × ℝ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M49">View MathML</a>

(6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M50">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M51">View MathML</a> if N > 4.

Choose ε > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M52">View MathML</a>. By (6), the Poincaré inequality and the Sobolev inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M53">View MathML</a>

So, part (a) holds if we choose ||u|| = ρ > 0 small enough.

On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M54">View MathML</a>, take ε > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M41">View MathML</a>. By (7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M55">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M41">View MathML</a> and ||ϕl|| = 1, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M56">View MathML</a>

and part (b) is proved.

Lemma 2.4. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M57">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M58">View MathML</a>. If f satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M8">View MathML</a>-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M59">View MathML</a>, then

(i) the functional I is coercive on W, that is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M60">View MathML</a>

and bounded from below on W,

(ii) the functional I is anti-coercive on V.

Proof. For u W, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a>, for any ε > 0, there exists Bl = Bl(ε) such that for all (x, s) ∈ Ω × ℝ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M61">View MathML</a>

(8)

So we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M62">View MathML</a>

Choose ε > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M63">View MathML</a>. This proves (i).

(ii) When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M64">View MathML</a>, it is easy to see that the conclusion holds.

Lemma 2.5. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M64">View MathML</a>, then I satisfies the (PS) condition.

Proof. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M65">View MathML</a> be a sequence such that |I (un)| ≤ c, < I'(un), ϕ > → 0.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M66">View MathML</a>

(9)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M38">View MathML</a>. If |un|2 is bounded, we can take ϕ = un. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a>, there exists a constant c > 0 such that |f (x, un(x))| ≤ c|un(x)|, a.e. x ∈ Ω. So (un) is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19">View MathML</a>. If |un|2 → +∞, as n → ∞, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M67">View MathML</a>, then |vn|2 = 1. Taking ϕ = vn in (9), it follows that ||vn|| is bounded. Without loss of generality, we assume vn v in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M19">View MathML</a>, then vn → v in L2(Ω). Hence, vn → v a.e. in Ω. Dividing both sides of (9) by |un|2, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M38">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M68">View MathML</a>

(10)

Then for a.e. x ∈ Ω and suitable ϵ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M69">View MathML</a>

as n → ∞. In fact, if v(x) ≠ 0, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M5">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M70">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M71">View MathML</a>

as n → ∞. If v(x) = 0, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M72">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M73">View MathML</a>, by choosing ϕ = v in (10) and the Lebesgue dominated convergence theorem, we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M74">View MathML</a>

From Fourier series theory, it is easy to see that v ≡ 0. It contradicts to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M75">View MathML</a>.

It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equation. Let us recall some results which will be used later. We refer the readers to the book [18] for more information on Morse theory.

Let H be a Hilbert space and I ∈ Cl(H, ℝ) be a functional satisfying the (PS) condition or (C) condition, and Hq (X, Y) be the qth singular relative homology group with integer coefficients. Let u0 be an isolated critical point of I with I(u0) = c, c ∈ ℝ, and U be a neighborhood of u0. The group

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M76">View MathML</a>

is said to be the qth critical group of I at u0, where Ic = {u H: I (u) ≤ c}.

Let K: = {u H: I'(u) = 0} be the set of critical points of I and a < inf I (K), the critical groups of I at infinity are formally defined by (see [19])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M77">View MathML</a>

The following result comes from [18,19] and will be used to prove the result in this article.

Proposition 2.6 [19]. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M78">View MathML</a>, I is bounded from below on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M79">View MathML</a>and I(u) → -as ||u|| with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M80">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M81">View MathML</a>

(11)

3 Proof of the main result

Proof of Theorem 2.1. By Lemmas 2.2 and 2.3 and the mountain pass theorem, the functional I+ has a critical point ul satisfying I+(ul) ≥ β. Since I+ (0) = 0, ul ≠ 0 and by the maximum principle, we get ul > 0. Hence ul is a positive solution of the problem (1) and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M82">View MathML</a>

(12)

Using the results in [18], we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M83">View MathML</a>

(13)

Similarly, we can obtain another negative critical point u2 of I satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M84">View MathML</a>

(14)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M27">View MathML</a>, the zero function is a local minimizer of I, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M85">View MathML</a>

(15)

On the other hand, by Lemmas 2.4, 2.5 and the Proposition 2.6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M86">View MathML</a>

(16)

Hence I has a critical point u3 satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/39/mathml/M87">View MathML</a>

(17)

Since k ≥ 2, it follows from (13)-(17) that ul, u2, and u3 are three different nontrivial solutions of the problem (1).

Competing interests

The author declares that he has no competing interests.

Acknowledgements

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

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