On Fourth-Order Elliptic Equations of Kirchhoff Type with Dependence on the Gradient and the Laplacian

Abstract

We consider a nonlocal fourth-order elliptic equation of Kirchhoff type with dependence on the gradient and Laplacian , in , , , on , where , are positive constants. We will show that there exists such that the problem has a nontrivial solution for through an iterative method based on the mountain pass lemma and truncation method developed by De Figueiredo et al., 2004.

1. Introduction

This paper concerns with the existence of solutions of the fourth-order Kirchhoff type elliptic equation as follows:where is a bounded and smooth domain in (), , are positive constants, and is locally Lipschitz continuous.

The fourth-order elliptic equation arises in the study of traveling waves in suspension bridges, which has been extensively investigated in recent years, such as [1–6]. To our attention, some authors paid more attention to a more general biharmonic elliptic problem For this problem, due to the presence of and in , it is not variational. To overcome this difficulty, in [5], Wang deals with this problem via the upper and lower solutions and monotone iterative methods; in [7], the authors apply a technique developed by De Figueiredo et al. [8, 9] in studying a second-order elliptic problem involving the gradient, which “freezes" the gradient, and use truncation on the nonlinearity . Thus the new problem becomes variational and an iterative scheme of the mountain pass “approximated” solutions is built.

In addition, the nonlocal fourth-order equation has also been studied by many authors. We refer the readers to [10–20]. Particularly, Wang et al. studied the following fourth-order equation of Kirchhoff type equation where is a positive parameter. The authors showed that there exists such that the fourth-order elliptic equation has a nontrivial solution for by using the mountain pass iterative techniques and the truncation method.

Motivated by these works, to study problem (1), we combine the famous mountain pass lemma with a technique developed by De Figueiredo et al. [8], which “freezes” the gradient and the Laplacian variable and use truncation on the nonlinearity of . For convenience, we recall a definition and restate the mountain pass theorem.

Definition 1. Let be a real Banach space and a -functional. A sequence in is a (PS)-sequence for if for some constant as , while as . We say that the functional satisfies the (PS)-condition if any (PS)-sequence for has a convergent subsequence.

Theorem A (mountain pass lemma). Let be a real Banach space; satisfying (PS)-condition. Suppose the following: (1)There exist , such that  where .(2)There is and such that Then has a critical value which can be characterized as where .

2. The Main Result

Theorem 2. Assume that the function satisfies the following conditions:
is locally Lipschitz continuous, and there exist ,   which satisfy such that for all .
uniformly with respect to , and .
There exist and such that where .
There exist positive constants () depending on , , , and such that () satisfy where and () are the optimal constants of the following inequalities:where is the norm of the Hilbert space defined by Then there exists such that (1) has at least a nontrivial solution for .

For each and , we study the following “truncate" and “freezed” problemwhere satisfies , and The associated functional is defined as where

Lemma 3. Let and be fixed. Then (1)there exist constants such that with ;(2)for fixed with , as

Proof. On one hand, by , for any , there exists a constant such that, for , one hasOn the other hand, if , from it follows that there exists such that Then, from (20), (21), and the Sobolev inequality, we have for some positive constant . Therefore, for sufficiently small , we can choose and such that the first result of Lemma 3 holds.
Now, we show that implies that there exist such thatIn fact, from , we have , for any . Being integral from to , we get namely, Then Let and then inequality (23) holds.
Taking an arbitrary with , then from (23), we get which implies that the second result of Lemma 3 holds.

Lemma 4. Let and be fixed. Then the functional satisfies the (PS)-condition.

Proof. Let be a (PS)-sequence; namely,From the standard processes, we only need to prove that is bounded in . On a contradiction, suppose that ; then, from , we obtain On the other hand, from (29) we know that Then, from the above inequalities, we get which contradicts with . Therefore the sequence is bounded in .

Lemma 5. For any and , problem (15) has a nontrivial weak solution.

Proof. By Theorem A, Lemmas 3, and 4, the result holds.

Lemma 6. Let be fixed. Then there exist positive constants and , independent of , such that for every solution obtained in Lemma 5.

Proof. Firstly, since , from (23) it follows that As , we can get a such that ; that is, Define , where is defined in . Then we get Furthermore, we have where is independence of , , and . Therefore, , for some .
Secondly, from and , given , there exists such that Since , it is easy to obtain that for some constants . Therefore, there exists such that .

Lemma 7 (see [7]). Let be fixed, and choose for . If is a weak solution of problem (15), then for some , and if .

Lemma 8. There exist three constants (), independent of , , and , such that In addition, there exists such that

Proof. From (37) and the proof of Lemma 6, there exists , independent of and , such that Then by Lemma 7 and the Sobolev embedding theorem, the inequalities in the lemma are as follows. In addition, since and , there exists a sufficiently large such that .

Now let () be the weak solution of the following problem:with , where was found in Lemma 5 and obtained in Lemma 8. From Lemmas 6–8, we have satisfying and Thus

Lemma 9. Assume that holds. Let Then strongly converges in .

Proof. Let and be the weak solutions of (43) with and , respectively. Then, multiplying (43) by , we obtain Furthermore, by , , and the Hlder inequality, we haveHence, by (47) and (48), we get namely, Now, choosing , then Therefore, converges strongly in .

Proof of Theorem 2. Firstly, from Lemma 6, we get , and , , are uniformly bounded. Secondly, set ; then Since , for some positive constant , by the Schauder theorem, there exists a constant such that ; that is, . Furthermore, by the Arzela-Ascoli theorem and Lemma 9, the sequence satisfies uniformly in for and . Finally, passing to the limit in (43), we obtain that is a classical solution of (1).

Example 10. Consider the following problem: where and and are positive and continuous functions. It is easy to verify that satisfies all the conditions of –.

3. Conclusion

The paper considers a class of fourth-order elliptic equations of Kirchhoff type with dependence on the gradient and Laplacian. The existence of a nontrivial solution of (1) is established when we choose appropriate such that . The paper generalized the conclusions in [7, 14] and weakened the condition in [7]. In the following research work, we will also consider problem (1), but we just truncate the right side of the equation, and the left of the equation remains the same.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally and significantly to writing this article. All the authors read and approved the final manuscript.

Acknowledgments

Yunhai Wang was supported by the projects of Guizhou Provincial Science and Technology Fund (QKH-JICHU 2017, Grant no. 1408) and the doctoral starting up foundation of Guizhou Institute of Technology (Grant no. XJGC20150408). Fanglei Wang was supported by NNSF of China (no. 11501165) and the Fundamental Research Funds for the Central Universities (2015B19414).