Research Article | Open Access
Qiong Liu, Dengfeng Lü, "Multiplicity of Solutions for a Class of Fourth-Order Elliptic Problems with Asymptotically Linear Term", Journal of Applied Mathematics, vol. 2012, Article ID 749059, 14 pages, 2012. https://doi.org/10.1155/2012/749059
Multiplicity of Solutions for a Class of Fourth-Order Elliptic Problems with Asymptotically Linear Term
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 , 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 [1–3]). 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 [4–14].
For problem (1.3), Lazer and McKenna  proved the existence of solutions when and ( is the sequence of the eigenvalues of in ) by the global bifurcation method. In , 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 , 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  by using variational method, and the existence of a nontrivial solution has been obtained in  by using the mountain pass theorem. For more similar problems, we refer to [10–20] 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 Cerami’s condition and the symmetric mountain pass lemma (see ), 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 –, 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 –, 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 .
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 Cerami’s condition as follows (see ).
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 .
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, , , , is compact, symmetric with respect to the origin, and for any , there holds . If , define
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 –. 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 –, 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)–(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 –, 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)–(3.26), Hölder 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 – 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 , 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.
The authors would like to thank the referees for carefully reading this paper and making valuable comments and suggestions.
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