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

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

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:

(1.1)

where is the biharmonic operator, and Ω is a bounded smooth domain in ().

The conditions imposed on are as follows:

(H1) , and there are constants such that

where ;

(H2) , , uniformly on Ω;

(H3) uniformly on Ω;

(H4) There is a constant such that for all and ,

where ;

(H5) For some , either

or

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 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 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

The function has a local linking at 0, with respect to if for some ,

(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 defined on a Hilbert space and , I has local linking at 0.

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

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

(a) bounded below [8],

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 be the eigenvalues of . Then () is the eigenvalue of , where . We assume that

(H6) , , uniformly in Ω, and .

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

(A1) , ;

(A2) , , for some,

(A3) , , for some,

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

### 2 Preliminaries

Let X be a Banach space with a direct sum decomposition

Consider two sequences of a subspace:

such that

For every multi-index , let . We know that

A sequence is admissible if for every , there is such that . For every , we denote by the function I restricted .

Definition 2.1 Let I be locally Lipschitz on X and . The functional I satisfies the condition if every sequence such that is admissible and

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

Definition 2.2 Let I be locally Lipschitz on X and . The functional I satisfies the condition if every sequence such that is admissible and

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

Remark 2.1

1. The condition implies the condition for every .

2. When the sequence is bounded, then the sequence is a sequence (see [11]).

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

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

The function has a local linking at 0, with respect to , if for some ,

Lemma 2.1 (see [10])

Suppose thatsatisfies the following assumptions:

(B1) Ihas a local linking at 0 and;

(B2) Isatisfies;

(B3) Imaps bounded sets into bounded sets;

(B4) for every, , , . ThenIhas at least two critical points.

Remark 2.2 Assume I satisfies the condition. Then this theorem still holds.

Let X be a real Hilbert space and let . The gradient of I has the form

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,

and two sequences of finite-dimensional subspaces,

such that

For every multi-index , we denote by the space

by the orthogonal projector onto , and by 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;

(ii) there exists a compact self-adjoint operatorsuch that

(iii) is invertible;

(iv) for infinitely many multiple-indices,

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

defined on . We consider only the case , and

(3.1)

Then other case is similar and simple.

Let be the finite dimensional space spanned by the eigenfunctions corresponding to negative eigenvalues of and let be its orthogonal complement in X. Choose a Hilbertian basis () for X and define

By the condition (H1) and Sobolev inequalities, it is easy to see that the functional I belongs to and maps bounded sets to bounded sets.

(2) We claim that I has a local linking at 0 with respect to . Decompose into when , . Also, set , , , . By the equivalence of norm in the finite-dimensional space, there exists such that

(3.2)

It follows from (H1) and (H2) that for any , there exists such that

(3.3)

Hence, we obtain

where , is a constant and hence, for small enough,

Let be such that and let

From (3.2), we have

for all and . On the one hand, one has for all . Hence, from (H5), we obtain

On the other hand, we have

for all . It follows from (3.3) that

for all and all with , which implies that

where is a constant. Hence, there exist positive constants , and such that

for all with , which implies that

for small enough.

(3) We claim that I satisfies . Consider a sequence such that is admissible and

(3.4)

and

(3.5)

Let . Up to a subsequence, we have

If , we choose a sequence such that

For any , let . By the Sobolev imbedded theory, we have

So, for n large enough, , and combining Ehrling-Nirenberg-Gagliardo inequality, we have

(3.6)

where ϵ is a small enough constant.

That is, . Now, , , we know that and

(3.7)

Therefore, using (H3), we have

If , then the set has a positive Lebesgue measure. For , we have . Hence, by (H3), we have

(3.8)

From (3.4), we obtain

(3.9)

By (3.8), the right-hand side of (3.9) . This is a contradiction.

In any case, we obtain a contradiction. Therefore, is bounded.

Finally, we claim that for every ,

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

So, for any , we have

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.

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