Nonlinear Functional Analysis of Boundary Value Problems 2013View this Special Issue
Existence and Multiplicity of Nontrivial Solutions for a Class of Fourth-Order Elliptic Equations
Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.
1. Introduction and Main Results
Consider the fourth-order Navier boundary value problem where is a bounded smooth domain, , and . is the Laplace operator and is the biharmonic operator.
Let be the eigenvalues of in . The eigenvalue problem has infinitely many eigenvalues .
We will always assume . Let be the Hilbert space . is equipped with the inner product and the norm
A weak solution of problem (1) is any such that for any .
Let be the functional defined by where . And, one has for any , so that a critical point of the functional in corresponds to a weak solution of problem (1).
In recent years, fourth-order problems have been studied by many authors. In , Lazer and McKenna have pointed out that problem (1) furnishes a model to study travelling waves in suspension bridges if , where and . Since then, more general nonlinear fourth-order elliptic boundary value problems have been studied.
In [2, 3], Micheletti and Pistoia proved that problem admits two or three solutions by variational method. In , Zhang obtained the existence of weak solutions for problem (8) when is sublinear at . In , Zhang and Li showed that problem (8) has at least two nontrivial solutions by means of Morse theory and local linking. When is asymptotically linear at infinity, the existence of three nontrivial solutions has been obtained in  by using Morse theory. In , by using the mountain pass theorem, An and Liu gave the existence result for nontrivial solutions for a class of asymptotically linear fourth-order elliptic equations. In , Zhou and Wu got the existence of four sign-changing solutions or infinitely many sign-changing solutions for (8) by using the sign-changing critical point theorems. In , Yang and Zhang showed new results on invariant sets of the gradient flows of the corresponding variational functionals and proved the existence of positive, negative, and sign-changing solutions for some fourth-order semilinear elliptic boundary value problems. In , by using the variational method, Liu and Huang obtained an existence result of sign-changing solutions as well as positive and negative solutions for a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity.
In this paper, we will study the existence of nontrivial solutions of problem (1). Our main results are the following theorems.
Theorem 1. Assume that is even in and the following conditions hold: () uniformly in ;()there exist two constants and such that uniformly for all ;()there exists a constant such that uniformly for all .Then problem (1) has infinitely many nontrivial solutions.
Theorem 2. Assume that satisfies () and ()there exist three positive constants , , and such that(), if ;(), if , where .If is even in , problem (1) has infinitely many nontrivial solutions.
Theorem 4. Assume that satisfies (), (), (), and () uniformly in .If is an eigenvalue of (with Navier boundary condition), assume also the condition that ()there exists such that(i), for all ; or(ii), for all . Then problem (1) has at least one nontrivial solution.
Theorem 5. Suppose that satisfies (), (), and (). If is an eigenvalue of (with Navier boundary condition), assuming also (), then problem (1) has at least one nontrivial solution.
Here, we have where , and is the space spanned by the eigenvectors corresponding to negative (positive) eigenvalues of . It is easy to know that under the conditions of our theorems.
It is well known that is continuously embedded in for every . If , the embedding is compact. It follows from (), (), and () that Hence, there is a positive constant such that for , , where denotes the norm of .
2. Proof of Main Results
In this paper, we will use the Fountain Theorem of Bartsch ([13, Theorem 2.5], [14, Theorem 3.6]) to prove our Theorems 1 and 2. And, we will prove Theorems 4 and 5 by using a version of Local Linking theorem [12, Theorem 2.2] which extends theorems given by Li and Willem , Li and Szulkin .
In [13, 14], Bartsch established the Fountain Theorem under the condition. Since the Deformation Theorem is still valid under the Cerami condition, the Fountain Theorem is true under the Cerami condition. So, we have the following Fountain Theorem.
Let be a reflexive and separable Banach space. It is well known that there exist such that(1) where for and for .(2), .
Let ; then . We define
Theorem A (Fountain theorem). Assume that satisfies the Cerami condition (), . If for almost every , there exist such that ()() then has an unbounded sequence of critical values.
For the reader's convenience, we state the following Local Linking theorem [12, Theorem 2.2]. Let be a real Banach space with and such that . For every multi-index , let . We define that . A sequence is admissible if for every there is such that . We say that satisfies the condition if every sequence such that is admissible and satisfies contains a subsequence which converges to a critical point of , where .
Theorem B (see [12, Theorem 2.2]). Suppose that satisfies the following assumptions: () and has a local linking at ; that is, for some , () satisfies condition;() maps bounded sets into bounded sets;()for every as , on . Then has at least one nonzero critical point.
Now, we can give the proof of our theorems.
Proof of Theorem 1. At first, we claim that satisfies the Cerami condition (). Consider a sequence such that is bounded and as . Then there exists a constant such that
By a standard argument, we only need to prove that is a bounded sequence in . Otherwise, going if necessary to a subsequence, we can assume that as . From (), there exist two constants such that
So, by (21) and (22), we have
which implies that
for all and some positive constant .
Since On the one hand, we consider the case
Putting one has . Let where . We can obtain from Hölder's inequality, (15), and (24) that for all , where .
By () and (29), one has for all . Since , we have
On the other hand, if satisfies, then one sees . So, we get It follows from (24) that By () and (33), we obtain for all . Note that and imply that . So, it follows from (34) and the above expression that Hence, we conclude from (31) and (36) that Similarly for , we get
It follows from the equivalence of the norms on the finite dimensional space that there exists such that Putting , one has It follows from (24), (39), and Hölder's inequality that and consequently Hence, by (37), (38), and (42), one sees as , which is a contradiction. So, we obtain that is bounded in . By a standard argument, we get that satisfies the condition .
Let with for any . Set Since , all the norms are equivalent. For , there exists a constant such that From condition (), there exists such that For , it follows from (45) and (46) that which implies that So, () of Theorem A is satisfied for every large enough.
Here, we obtain from () that there exists a positive constant such that uniformly for all . Let us define For large enough, one has . By (49), on , we have Choosing , we obtain, if and , Since, by Lemma 3.8 of , as , () is proved. Hence, the proof is completed by using Fountain theorem.
Proof of Theorem 2. Firstly, we claim that satisfies the Cerami condition (). Consider a sequence such that is bounded from above and as . By a standard argument, we only need to prove that is a bounded sequence in . For otherwise, we can assume that as .
From assumption (), there exist two positive constants and , such that So, one has for all and some positive constant .
Let ; then and for all . By (54), we have as . So, for , it follows from Hölder's inequality and the above expression that as . It follows from (39) that Hence, we get From (), (53), and (56), there exists a positive constant such that as ; therefore, , which is a contradiction. Hence, is bounded.
In a way similar to the proof of Theorem 1, we can obtain that satisfies () of Theorem A.
It follows from () that there is such that Then, by () and (60), for , one has which implies that for .
Therefore, there exist two positive constants and such that where . As the proof of Theorem 1, we can get (). Therefore, Theorem 2 holds.
Proof of Theorem 4. The proof of this theorem is divided in several steps.
Step 1. We claim that has a local linking at zero with respect to .
By (), for any , there exists such that We obtain from the above expression and (49) that where . Hence, we can get from (15) and (65) that for all .
Here, we consider only the case where is an eigenvalue of and case (ii) of () holds. The case (i) is similar.
Let , and , where . Choose a Hilbertian basis for and define Now, by (66), for each , one has Letting and by , we have for small enough.
Let be such that . Put Then, for all and , by (39), one sees On one hand, from above expression, for any , we have Hence, by condition (ii) of (), we get On the other hand, for any , one has Hence, for all and with , we can obtain from (65) that which implies that Letting in above expression, then for all and with , we have which implies that for small enough. Hence, has a local linking at zero with respect to for small enough.
Step 2. In a way similar to the proof of Theorem 1, we can get that satisfies the condition.
Step 3. Now, we claim that for each , one has Since and , all the norms are equivalent. For , there exists a constant such that From condition (), there exists a constant such that For , it follows from (80) and (81) that which implies that Hence, all the assumptions of Theorem B are verified. Then, the proof of Theorem 4 is completed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referee for the valuable suggestions. This paper is supported by the National Natural Science Foundation of China (no. 11071198) and the Fundamental Research Funds for the Central Universities (no. XDJK2010C055).