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Existence and multiplicity of solutions for a fourth-order elliptic equation

Fanglei Wang1* and Yukun An2

Author affiliations

1 College of Science, Hohai University, Nanjing, 210098, P. R. China

2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China

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Citation and License

Boundary Value Problems 2012, 2012:6  doi:10.1186/1687-2770-2012-6


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/6


Received:26 August 2011
Accepted:17 January 2012
Published:17 January 2012

© 2012 Wang and An; 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

This article is concerned with the existence and multiplicity of nontrival solutions for a fourth-order elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M1">View MathML</a>

by using the mountain pass theorem.

Keywords:
fourth-order elliptic equation; nontrivial solutions; mountain pass theorem

1 Introduction

In this article we study the existence of nontrivial solutions for the fourth-order boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M2">View MathML</a>

(1)

where Ω ⊂ RN is a bounded smooth domain, f : Ω × R R and M : R R are continuous functions. The existence and multiplicity results for Equation (1) are considered in [1-3] by using variational methods and fixed point theorems in cones of ordered Banach space with space dimension is one.

On the other hand, The four-order semilinear elliptic problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M3">View MathML</a>

(2)

arises in the study of traveling waves in a suspension bridge, or the study of the static deflection of an elastic plate in a fluid, and has been studied by many authors, see [4-10] and the references therein.

Inspired by the above references, the object of this article is to study existence and multiplicity of nontrivial solution of a fourth-order elliptic equation under some conditions on the function M(t) and the nonlinearity. The proof is based on the mountain pass theorem, namely,

Lemma 1.1. Let E be a real Banach space, and I C1(E, R) satisfy (PS)-condition. Suppose

(1) There exist ρ > 0, α > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M4">View MathML</a>

where Bp = {u E|∥u∥ ≤ ρ}.

(2) There is an e E and ∥e∥ > ρ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M5">View MathML</a>

Then I(u) has a critical value c which can be characterized as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M6">View MathML</a>

where Γ = {γ C([0, 1],E)|γ(0) = 0,γ(1) = e}.

The article is organized as follows: Section 2 is devoted to giving the main result and proving the existence of nontrivial solution of Equation (1). In Section 3, we deal with the multiplicity results of Equation (1) whose nonlinear term is asymptotically linear at both zero and infinity

2 Main result I

Theorem 2.1. Assume the function M(t) and the nonlinearity f(x, t) satisfying the following conditions:

(H1) M(t) is continuous and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M7">View MathML</a>

(3)

for some m0 > 0. In addition, that there exist m' > m0 and t0 > 0, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M8">View MathML</a>

(4)

(H2) f(x, t) ∈ C(Ω × R); f(x, t) ≡ 0, ∀x ∈ Ω, t ≤ 0, f(x, t) ≥ 0, ∀x ∈ Ω, t > 0;

(H3) |f(x, t)| ≤ a(x) + b|t|p, ∀t R and a.e. x in Ω, where a(x) ∈ Lq (Ω), b R

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M9">View MathML</a> if N > 4 and 1 < p < ∞ if N ≤ 4 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M10">View MathML</a>;

(H4) f(x, t) = o(|t|) as t → 0 uniformly for x ∈ Ω ;

(H5) There exists a constant Θ > 2 and R > 0, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M11">View MathML</a>

Then Equation (1) has at least one nonnegative solution.

Let Ω ⊂ RN be a bounded smooth open domain, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M12">View MathML</a> be the Hilbert space equipped with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M13">View MathML</a>

and the deduced norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M14">View MathML</a>

Let λ1 be the positive first eigenvalue of the following second eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M15">View MathML</a>

Then from [4], it is clear to see that Λ1 = λ1(λ1 - c) is the positive first eigenvalue of the following fourth-order eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M16">View MathML</a>

where c < λ1. By Poincare inequality, for all u H, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M17">View MathML</a>

(5)

A function u H is called a weak solution of Equation (1) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M18">View MathML</a>

holds for any v H. In addition, we see that weak solutions of Equation (1) are critical points of the functional I : H R defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M19">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M20">View MathML</a> and F(x, t) = ∫ f(x, t)dt. Since M is continuous and f has subcritical growth, the above functional is of class C1 in H. We shall apply the famous mountain pass theorem to show the existence of a nontrivial critical point of functional I(u).

Lemma 2.2. Assume that (H1)-(H5) hold, then I(u) satisfies the (PS)-condition.

Proof. Let {un} ⊂ H be a (PS)-sequence. In particular, {un} satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M21">View MathML</a>

(6)

Since f(x, t) is sub-critical by (H3), from the compactness of Sobolev embedding and, following the standard processes we know that to show that I verifies (PS)-condition it is enough to prove that {un} is bounded in H. By contradiction, assume that ∥un∥ → +∞.

Case I. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M22">View MathML</a> is bounded, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M23">View MathML</a>. We assume that there exist a constant K > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M24">View MathML</a>. By (H1), it is easy to obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M25">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M26">View MathML</a>. Then, from

(H1), (H3), and (H5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M27">View MathML</a>

On the other hand, it is easy to obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M28">View MathML</a>

Then, from above, we can have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M29">View MathML</a>

which contradicts ∥un∥ → +∞. Therefore {un} is bounded in H.

Case II. if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M23">View MathML</a>. By (H1), let l2 = max{1, m'}, we also can obtain that {un} is bounded in H.

This lemma is completely proved.

Lemma 2.3. Suppose that (H1)-(H5) hold, then we have

(1) there exist constants ρ > 0, α > 0 such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M30">View MathML</a> with Bp = {u H u∥ ≤ ρ};

(2) I(1) → -∞ as t → +∞.

Proof. By (H1)-(H4), we see that for any ε > 0, there exist constants C 1 > 0, C2 such that for all (x, s) ∈ Ω × R, one have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M31">View MathML</a>

(7)

Choosing ε > 0 small enough, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M32">View MathML</a>

by (3), (5), (7) and the Sobolev inequality. So, part 1 is proved if we choose ∥uρ > 0 small enough.

On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M33">View MathML</a>

using (4) and (H5). Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M34">View MathML</a>

as t → +∞ and part 2 is proved.

Proof of Theorem 2.1. From Lemmas 2.2 and 2.3, it is clear to see that I(u) satisfies the hypotheses of Lemma 1.1. Therefore I(u) has a critical point.

3 Existence result II

Theorem 3.1. Assume that (H1) holds. In addition, assume the following conditions are hold:

(H6) f(x, t)t ≥ 0 for x ∈ Ω, t R;

(H7) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M35">View MathML</a>, uniformly in a.e x ∈ Ω, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M36">View MathML</a>.

Then Equation (1) has at least two nontrivial solutions, one of which is positive and the other is negative.

Let u+ = max{u, 0}, u- = min{u, 0}. Consider the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M37">View MathML</a>

(8)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M38">View MathML</a>

Define the corresponding functional I+ : H R as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M39">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M40">View MathML</a>. Obviously, I+ Cl(H, R). Let u be a critical point of I+ which implies that u is the weak solution of Equation (8). Futhermore, by the weak maximum principle it follows that u ≥ 0 in Ω. Thus u is also a solution of Equation (1).

Similarly, we also can define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M41">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M42">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M43">View MathML</a>. Obviously, I- C1(H, R). Let u be a critical point of I- which implies that u is the weak solution of Equation (1) with I-(u) = I(u).

Lemma 3.2. Assume that (H1), (H6), and (H7) hold, then I± satisfies the (PS) condition.

Proof. We just prove the case of I+. The arguments for the case of I- are similar. Since Ω is bounded and (H7) holds, then if {un} is bounded in H, by using the Sobolve embedding and the standard procedures, we can get a convergent subsequence. So we need only to show that {un} is bounded in H.

Let {un} ⊂ H be a sequence such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M44">View MathML</a>

(9)

By (H7), it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M45">View MathML</a>

Now, (9) implies that, for all ϕ H, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M46">View MathML</a>

(10)

Set ϕ = un, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M47">View MathML</a>

(11)

Next, we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M48">View MathML</a> is bounded. If not, we may assume that ∥unL2 → +∞ as n → +∞. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M49">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M50">View MathML</a>. From (11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M51">View MathML</a>

thus {ωn} is bounded in H. Passing to a subsequence, we may assume that there exists ω H with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M52">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M53">View MathML</a>

On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M54">View MathML</a> as n → +∞, by Poincare inequality, it is easy to know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M23">View MathML</a> as n → +∞. Thus by (H1), the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M55">View MathML</a>. So as n → +∞, by (10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M56">View MathML</a>

(12)

Then ω H is a weak solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M57">View MathML</a>

The weak maximum principle implies that ω = ω+ ≥ 0. Choosing ϕ (x) = φ1(x) > 0, which is the corresponding eigenfunctions of λ1. From (10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M58">View MathML</a>

(13)

On the other hand, we can easily see that Λ = λ1(λ1 + m') is the eigenvalue of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M59">View MathML</a>

and the corresponding eigenfunction is still φ1(x). If ω(x) > 0, we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M60">View MathML</a>

(14)

which follows that ω ≡ 0 by Λ < β But this conclusion contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M52">View MathML</a>.

Hence {un} is bounded in H.

Now we prove that the functionals I± has a mountain pass geometry.

Lemma 3.3. Assume that (H1), (H7) hold, then we have

(1) there exists ρ, R > 0 such that I±(u) > R, if ∥u∥ = ρ;

(2) I±(u) are unbounded from below.

Proof. By (H7), for any ε > 0, there exists C 1 > 0, C2 > 0 such that ∀(x, s) ∈ Ω × R, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M61">View MathML</a>

(15)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M62">View MathML</a>

(16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M63">View MathML</a>

We just prove the case of I+. The arguments for the case of I- are similar. Let ϕ = 1. When t is sufficiently large, by (16) and (H1), it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M64">View MathML</a>

On the other hand, by (17), (H1), the Poincare inequality and the Sobolve embedding, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/6/mathml/M65">View MathML</a>

where C4 is a constant. Choosing ∥u∥ = ρ small enough, we can obtain I+(u) ≥ R > 0 if ∥u∥ = ρ.

Proof of Theorem 3.1. From Lemma 3.3, it is easy to see that there exists e H with ∥e∥ > ρ such that I±(e) < 0.

Define

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and

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From Lemma 3.3, we have

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Moreover, by Lemma 3.2, the functions I± satisfies the (PS)-condition. By Lemma 1.1, we know that c+ is a critical value of I+ and there is at least one nontrivial critical point in H corresponding to this value. This critical in nonnegative, then the strong maximum principle implies that is a positive solution of Equation (1). By an analogous way we know there exists at least one negative solution, which is a nontrivial critical point of I- Hence, Equation (1) admits at least a positive solution and a negative solution.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

In this manuscript the authors studied the existence and multiplicity of solutions for an interesting fourth-order elliptic equation by using the famous mountain pass lemma. Moreover, in this work, the authors' supplements done in [1-3]. All authors typed, read and approved the final manuscript.

Acknowledgements

The authors' would like to thank the referees for valuable comments and suggestions for improving this article.

References

  1. Ma, TF: Existence results for a model of nonlinear beam on elastic bearings. Appl. Math Lett. 13, 11–15 (2000). Publisher Full Text OpenURL

  2. Ma, TF: Existence results and numerical solutions for a beam equation with nonlinear boundary conditions. Appl Numer Math. 47, 189–196 (2003). Publisher Full Text OpenURL

  3. Ma, TF: Positive solutions for a nonlocal fourth-order equations of Kirchhoff type. Discrete Contin Dyn Syst (suppl). 694–703 (2007)

  4. An, Y, Liu, R: Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equations. Nonlinear Anal. 68, 3325–3331 (2008). Publisher Full Text OpenURL

  5. Bernis, F, Azorero, JG, Peral, I: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth-order. Adv Diff Equ. 1, 219–240 (1996)

  6. Chen, Y, McKenna, PJ: Traveling waves in a nonlinear suspension beam: the-oretical results and numerical observations. J Diff Equ. 135, 325–355 (1997)

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

  8. Lazer, AC, McKenna, PJ: Large-amplitude periodic oscillations in suspension bridge: some new connections with nonlinear analysis. SIAM Rev. 32, 537–578 (1990). Publisher Full Text OpenURL

  9. McKenna, PJ, Walter, W: Traveling waves in a suspension bridge. SIAM J Appl Math. 50, 703–715 (1990). Publisher Full Text OpenURL

  10. Pei, R: Multiple solutions for biharmonic equations with asymptotically linear nonlinearities. Bound Value Probl 2010, Article ID 241518. 11, (2010)