Journal of Applied Mathematics

VolumeΒ 2012, Article IDΒ 749059, 14 pages

http://dx.doi.org/10.1155/2012/749059

## Multiplicity of Solutions for a Class of Fourth-Order Elliptic Problems with Asymptotically Linear Term

School of Mathematics and Statistics, Hubei Engineering University, Hubei, Xiaogan 432000, China

Received 10 January 2012; Accepted 5 April 2012

Academic Editor: TurgutΒ ΓziΕ

Copyright Β© 2012 Qiong Liu and Dengfeng LΓΌ. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the following fourth-order elliptic equations: , where is a bounded domain with smooth boundary and is asymptotically linear with respect to at infinity. Using an equivalent version of Cerami's condition and the symmetric mountain pass lemma, we obtain the existence of multiple solutions for the equations.

#### 1. Introduction and Main Results

In this paper, we will investigate the existence of multiple solutions to the following fourth-order elliptic boundary value problem: where is a bounded domain with smooth boundary , is the biharmonic operator, ( is the first eigenvalue of in ) is a parameter. We assume that satisfies the following hypotheses.. uniformly for . uniformly for , where is a constant, or , and there exists , such that where . is odd in . uniformly for , where. is nondecreasing with respect to , for a.e. .

Problem (1.1) is usually used to describe some phenomena appeared in different physical, engineering and other sciences. In recent years, there are many results for the fourth-order elliptic equations. In [1], Lazer and McKenna considered the fourth-order problem: where and . They pointed out that this type of nonlinearity furnishes a model to study traveling waves in suspension bridges. They also presented a mathematical model for the bridge that takes account of the fact that the coupling provided by the stays connecting the suspension cable to the deck of the road bed is fundamentally nonlinear (see [1β3]). Since then, more general nonlinear fourth-order elliptic boundary value problems have been studied. Problem (1.1) and (1.3) have been studied extensively in recent years, we refer the reader to [4β14].

For problem (1.3), Lazer and McKenna [2] proved the existence of solutions when and ( is the sequence of the eigenvalues of in ) by the global bifurcation method. In [4], Tarantello found a negative solution when by a degree argument. For Problem (1.1), when , the existence of two or three nontrivial solutions has been obtained in [5, 6] for under certain conditions by using variational methods. In [7], positive solutions of problem (1.1) were got when satisfies the local superlinearity and sublinearity. When is asymptotically linear at infinity, the existence of three nontrivial solutions has been obtained in [8] by using variational method, and the existence of a nontrivial solution has been obtained in [9] by using the mountain pass theorem. For more similar problems, we refer to [10β20] and the references therein.

In this paper, we prove a new existence result about a multiple solutions of problem (1.1) under the assumption that is asymptotically linear with respect to at infinity. In this case, the Ambrosetti-Rabinowitz condition ((AR) condition for short) does not hold, hence it is difficult to verify the classical condition. To overcome this difficulty, by using an equivalent version of Ceramiβs condition and the symmetric mountain pass lemma (see [21]), we obtain the existence of multiple solutions for problem (1.1). To the best of our knowledge, our main results are new. Before stating the main results, we give some notations.

Set , then is a Hilbert space with the following inner product and the norm:

The corresponding energy functional of problem (1.1) is defined on by where . From β, it is easy to see that , it is well known that the weak solutions of problem (1.1) are the critical points of the energy functional .

Our main results are stated as follows.

Theorem 1.1. *Assume that satisfies assumptions β, and is given by (2.8). Then the following hold. *(i)*If is not an eigenvalue of problem (2.4), then problem (1.1) has at least pairs of nontrivial solutions in .*(ii)*Suppose that is satisfied, then the conclusion of (i) holds even if is an eigenvalue of problem (2.4).*(iii)*If , and holds, then problem (1.1) has infinitely many nontrivial solutions in .*

#### 2. Preliminaries

In this section, we give some preliminary results which will be used to prove our main results.

Throughout this paper, we will denote by the Lebesgue measure of , . will denote various positive constants, (respectively) denotes strong (respectively weak) convergence. denote as . denote Lebesgue spaces, the norm is denoted by for . The dual space of a Banach space will be denoted by .

First, we recall an equivalent version of Ceramiβs condition as follows (see [22]).

*Definition 2.1. *Let be a Banach space. is said to satisfy condition at level ( for short), if the following fact is true: any sequence , which satisfies
possesses a convergent subsequence in .

Next, we will state an abstract symmetric mountain pass lemma. For this purpose, we should first introduce the definition of genus (see [23β25]).

*Definition 2.2. *Let be a real Banach space and a subset of . is said to be symmetric if implies . For a closed symmetric set which does not contain the origin, we define a genus of by the smallest integer such that there exists an odd continuous mapping from to . If there does not exist such a , we define . Moreover, we set .

Let be an infinite dimensional real Banach space, β, , , is compact, symmetric with respect to the origin, and for any , there holds . If , define

Now, we recall an abstract symmetric mountain pass lemma, which can be found in [26, 27].

Lemma 2.3. *Let be linearly independent in , and , . Suppose that satisfies , , and condition for . Furthermore, there exists , such that in and . Then, if is bounded, then and is a critical value of . Moreover, if is bounded for all , and**
then , where . If is bounded for all , then possesses infinitely many critical values.*

Let us consider the eigenvalue problem:Set For , and are well defined. Furthermore, , and a real value is an eigenvalue of problem (2.4) if and only if there exists such that . At this point, let us set Then and is a manifold in . It follows from the standard Lagrange multiples arguments that eigenvalues of (2.4) correspond to critical values of , and satisfies the (PS) condition on . Thus a sequence of critical values of comes from the Ljusternik-Schnirelmann critical point theory on manifolds. For any , set Then values: are critical values and hence are eigenvalues of problem (2.4). Moreover, .

We prove some properties of functional in the following lemma.

Lemma 2.4. *For the functional defined by (1.5), if assumptions and hold, and for any with as , then there is a subsequence, still denoted by , such that**
holds for all , .*

*Proof. *This lemma is essentially due to [27, 28]. For the sake of completeness, we prove it here.

By as , for a suitable subsequence, we may assume that
We claim that for any and ,
Indeed, for any , at fixed and , we set
then
hence
Therefore,
and our claim (2.11) is proved.

On the other hand,
that is,
Combining (2.11) and (2.17) we have that

The proof is completed.

#### 3. Proof of the Main Results

We begin with the following lemma.

Lemma 3.1. *Let . Assume that satisfies assumptions β. Then the following hold. *(i)* satisfies condition if in assumption , and is not an eigenvalue of problem (2.4).*(ii)*If is an eigenvalue of problem (2.4) and holds, then satisfies condition.*(iii)*If , and holds, then satisfies condition.*

*Proof. *Suppose that is a sequence, that is, as , we have
It is easy to see that (3.2) implies that as , there hold
By Sobolev compact embedding, to show that satisfies condition, it suffices to show the boundedness of sequence in for each case.

(i) Suppose that and is not an eigenvalue of problem (2.4). Arguing by contradiction, we suppose that there exists a subsequence, still denoted by , such that as , there holds . Define
Then from assumptions β, there exists such that
Let
Obviously, is bounded in . Going if necessary to a subsequence, we can assume that
It is easy to show that . In fact, if , then from (3.3), (3.6), (3.8) and the definitions of and , as , we have
which is a contradiction.

From (3.6), there exists with such that, up to a subsequence, as , there holds
Then from (3.8) it follows that
On the other hand, from (3.3), (3.4), (3.5), and (3.7), we have
It follows from (3.11)β(3.13) that
Therefore (3.14) implies that satisfies
Let
Then as if , and as if . From assumption , for all . Thus (3.15) implies that satisfies
Therefore
This means that is an eigenvalue of problem (2.4), which contradicts our assumption, so is bounded in .

(ii) Suppose is an eigenvalue of problem (2.4), we need the additional assumption .

From assumption , there exists such that
and there exists such that
Furthermore, under assumptions β, there exists such that
Let , where is given by (3.21), is the best Sobolev constant such that
From assumption , there exists such that
For the above and each , set
From estimates (3.20), (3.1), (3.3), and (3.23), we get
where denotes the measure of .

On the other hand, for any fixed , from (3.1) and (3.3), we have
Since is bounded and , there exists a constant such that
Then, from (3.21)β(3.26), HΓΆlder inequality and Sobolev inequality, we have
that is, is bounded in .

(iii) Finally, we prove the case . Here the subcritical condition (1.2) is assumer as usual, but to make use of Lemma 2.4, is required in this case. Set
Then and is bounded in . Hence, up to a subsequence, we may assume that: there exists such that (3.8) also holds in this case. If , we claim that
In fact, if in , then (3.29) and (3.8) imply that
However, applying Lemma 2.4 with , we have
which contradicts (3.31), thus (3.30) holds.

On the other hand, similar to case (i), (3.13) holds. Let . Then by (3.30). From assumptions and , and as in , where is defined by (3.5). Hence, from (3.8) and (3.13), we have
which is a contradiction, thus , that is, up to a subsequence, is bounded in .

*Proof of Theorem 1.1. *The proof of this theorem is divided in two steps.*Step 1. *There exists , such that in and .

In fact, in each case, assumptions β imply that for any , there exists such that, for all , there holds
where is the same as that in (1.2), from which, it is easy to see that there exists ,β such that in and .*Step 2. *By the Symmetric Mountain Pass Lemma 2.3, to prove Theorem 1.1, it suffices to prove that for any , there exists a -dimensional subspace of and such that
First, we prove (3.35) in the case . Since , there is such that . By the definition of , there exists a -dimensional subspace of such that, for the above , there holds
that is,
By assumption , we have
Then, for the above , there exists large enough such that
Therefore, if with , by (3.39) and (3.37), we obtain
if and large enough.

If , similar to (3.37), for any , there exists such that
similar to (3.39), from assumption with it follows that there exists such that
Then, if with , we have
if and large enough. This completes the proof of Theorem 1.1.

#### Acknowledgment

The authors would like to thank the referees for carefully reading this paper and making valuable comments and suggestions.

#### References

- A. C. Lazer and P. J. McKenna, βLarge-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis,β
*SIAM Review*, vol. 32, no. 4, pp. 537β578, 1990. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. C. Lazer and P. J. McKenna, βGlobal bifurcation and a theorem of Tarantello,β
*Journal of Mathematical Analysis and Applications*, vol. 181, no. 3, pp. 648β655, 1994. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - P. J. McKenna and W. Reichel, βRadial solutions of singular nonlinear biharmonic equations and applications to conformal geometry,β
*Electronic Journal of Differential Equations*, vol. 37, pp. 1β13, 2003. View at Google Scholar Β· View at Zentralblatt MATH - G. Tarantello, βA note on a semilinear elliptic problem,β
*Differential and Integral Equations*, vol. 5, no. 3, pp. 561β565, 1992. View at Google Scholar Β· View at Zentralblatt MATH - A. M. Micheletti and A. Pistoia, βMultiplicity results for a fourth-order semilinear elliptic problem,β
*Nonlinear Analysis*, vol. 31, no. 7, pp. 895β908, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. M. Micheletti and A. Pistoia, βNontrivial solutions for some fourth order semilinear elliptic problems,β
*Nonlinear Analysis*, vol. 34, no. 4, pp. 509β523, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - G. Xu and J. Zhang, βExistence results for some fourth-order nonlinear elliptic problems of local superlinearity and sublinearity,β
*Journal of Mathematical Analysis and Applications*, vol. 281, no. 2, pp. 633β640, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - X. Liu and Y. Huang, βOn sign-changing solution for a fourth-order asymptotically linear elliptic problem,β
*Nonlinear Analysis*, vol. 72, no. 5, pp. 2271β2276, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. An and R. Liu, βExistence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation,β
*Nonlinear Analysis*, vol. 68, no. 11, pp. 3325β3331, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. Yang and J. Zhang, βExistence of solutions for some fourth-order nonlinear elliptic problems,β
*Journal of Mathematical Analysis and Applications*, vol. 351, no. 1, pp. 128β137, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Zhang and S. Li, βMultiple nontrivial solutions for some fourth-order semilinear elliptic problems,β
*Nonlinear Analysis*, vol. 60, no. 2, pp. 221β230, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Z. Jihui, βExistence results for some fourth-order nonlinear elliptic problems,β
*Nonlinear Analysis*, vol. 45, pp. 29β36, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Zhou and X. Wu, βSign-changing solutions for some fourth-order nonlinear elliptic problems,β
*Journal of Mathematical Analysis and Applications*, vol. 342, no. 1, pp. 542β558, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Zhang and Z. Wei, βMultiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin,β
*Journal of Mathematical Analysis and Applications*, vol. 383, no. 2, pp. 291β306, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - Y. Liu and Z. Wang, βBiharmonic equations with asymptotically linear nonlinearities,β
*Acta Mathematica Scientia. Series B*, vol. 27, no. 3, pp. 549β560, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. Lü, βMultiple solutions for a class of biharmonic elliptic systems with Sobolev critical exponent,β
*Nonlinear Analysis*, vol. 74, no. 17, pp. 6371β6382, 2011. View at Publisher Β· View at Google Scholar - Y. Yin and X. Wu, βHigh energy solutions and nontrivial solutions for fourth-order elliptic equations,β
*Journal of Mathematical Analysis and Applications*, vol. 375, no. 2, pp. 699β705, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - P. Mironescu and V. D. Rădulescu, βThe study of a bifurcation problem associated to an asymptotically linear function,β
*Nonlinear Analysis*, vol. 26, no. 4, pp. 857β875, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - P. Pucci and V. Rădulescu, βThe impact of the mountain pass theory in nonlinear analysis: a mathematical survey,β
*Bollettino della Unione Matematica Italiana. Serie 9*, vol. 3, no. 3, pp. 543β582, 2010. View at Google Scholar Β· View at Zentralblatt MATH - V. D. Rădulescu,
*Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic, and Variational Methods*, vol. 6 of*Contemporary Mathematics and Its Applications*, Hindawi Publishing Corporation, New York, NY, USA, 2008. - P. H. Rabinowitz,
*Minimax Methods in Critical Point Theory with Applications to Differential Equations*, vol. 65 of*CBMS Regional Conference Series in Mathematics*, Conference Board of the Mathematical Sciences, Washington, DC, USA, 1986. - D. G. Costa and C. A. Magalhães, βExistence results for perturbations of the $p$-Laplacian,β
*Nonlinear Analysis*, vol. 24, no. 3, pp. 409β418, 1995. View at Publisher Β· View at Google Scholar - J. Mawhin and M. Willem,
*Critical Point Theory and Hamiltonian Systems*, vol. 74 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 1989. - P. Mironescu and V. D. Rădulescu, βA multiplicity theorem for locally Lipschitz periodic functionals,β
*Journal of Mathematical Analysis and Applications*, vol. 195, no. 3, pp. 621β637, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Ambrosetti and P. H. Rabinowitz, βDual variational methods in critical point theory and applications,β vol. 14, pp. 349β381, 1973. View at Google Scholar Β· View at Zentralblatt MATH
- G. Li and H.-S. Zhou, βMultiple solutions to $p$-Laplacian problems with asymptotic nonlinearity as ${u}^{p-1}$ at infinity,β
*Journal of the London Mathematical Society. Second Series*, vol. 65, no. 1, pp. 123β138, 2002. View at Publisher Β· View at Google Scholar - G. Li and H.-S. Zhou, βAsymptotically linear Dirichlet problem for the $p$-Laplacian,β
*Nonlinear Analysis*, vol. 43, pp. 1043β1055, 2001. View at Publisher Β· View at Google Scholar - C. A. Stuart and H. S. Zhou, βApplying the mountain pass theorem to an asymptotically linear elliptic equation on ${R}^{N}$,β
*Communications in Partial Differential Equations*, vol. 24, no. 9-10, pp. 1731β1758, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH