Sign-Changing Solutions for Biharmonic Equations with Navier Boundary Conditions

Abstract

This work is concerned with sign-changing solutions to the biharmonic equation with Navier boundary conditions. By using the method of invariant sets of descending flow and symmetric mountain pass lemma, we obtain the existence of multiple sign-changing solutions when the nonlinear term is odd.

1. Introduction

In this paper, we deal with the following biharmonic equation with Navier boundary conditionswhere is a smooth bounded domain in .

The weak form of (1) is to look for satisfyingwhere is equipped with the norm

The equations involving the biharmonic operator have received special attention from many researchers, in part because they describe the traveling waves in a suspension bridge. Moreover, we cannot apply the maximum principle for the biharmonic operator for a general smooth bounded domain involving the Dirichlet boundary conditions, and this fact makes some problems involving the biharmonic operator very interesting from a mathematical point of view.

In the last decades, many authors have attached their attention to the existence and multiplicity of nontrivial solutions for biharmonic equations [1–7]. Using the dual method [2], establish the existence of a nodal ground state solution with for (1). When satisfies the asymptotically linear condition, An and Liu [1] proved the existence of nontrivial solutions for (1), their tool is the mountain pass theorem, which is usually used to get the nontrivial solution, see, for example, [8]. For the results of multiple nontrivial solutions to problem (1), see [9–12]. If is odd in and satisfies some additional conditions, by using the sign-changing critical theorems, the authors in [13] obtained that problem (1) admitted infinitely many sign-changing solutions [14] and established the existence and multiplicity of solutions without the AR condition.

In the case of the Laplacian operator, the study of the existence of nodal solutions has rich literature. However, we cannot use or adapt some techniques developed for the Laplacian, because in general, the authors prove the existence of sign-changing solutions by minimizing the energy functional on the set:

In problems involving the biharmonic operator, we cannot even ensure that given , one also has . In this work, the sign-changing solutions for (1) were obtained by the method of invariant sets of descending flows, and one of the key points is to construct a vector field , which keeps the positive and negative cones invariant. Consequently, constructing operator A (see Section 3 for details) and verifying its properties need special handling.

In what follows, we assume that satisfies the following conditions: , , ;  uniformly in ;  uniformly in , where ;  there exist and such that for a.e. , where . Here are the main results of the paper.

Theorem 1. Assume hold. Then, the problem (1) has a sign-changing solution. Moreover, if is odd in , then, the problem (1) has a sequence of sign-changing solutions.

2. Verification of the PS Condition

Now, we assume that satisfies – and define a functional on by

Clearly, is of class and

Lemma 1. I satisfies the (PS) condition.

Proof. Let be a (PS) sequence, thenwhere . Hence, is bounded in . Assume in . By Sobolev’s imbedding theorem, one has in .
Let be a ball with a radius and centered at 0. Define a cutoff function , such that if ; if ; , . Take as a test function, combining with Hölder inequality, we obtain thatTherefore, for all , one hasFirst, we verify that the weak limit satisfies equation (2), that is, is a weak solution.
Let and take as a test function, we obtainWithout losing generality, we assume for . By the local convergence (8), thenFor a general functional , take as a test function, where is the cutoff function defined before, thenLet ; thus, the Lebesgue dominant convergence theorem implies thatClearly,Once again, the Lebesgue dominant convergence theorem implies thatSimilarly,Altogether by letting in equation (13), equation (2) holds for all . By a further approximation, equation (2) holds for all , that is, is a weak solution. In particular, we haveAlso, one obtains thatCombining equation (18) with equation (19), thenNext, we show that . In order to get the results, we need the equality below.In fact, for , by the local convergence (5), we obtainOn the ball ,Hence,(21) follows from (22) and (24). Now, from (20) and (21), we getthat is, .

3. Properties of the Operator A

In this section, we define an operator , which is important to construct descending flow for the functional . is defined by

The weak form of equation (26) is

Then,

Lemma 2. The operator is well-defined and continuous.

Proof. Given , definethen is coercive and strictly convex, hence has a unique minimizer . The operator maps bounded set into bounded set. Let , , by equation (27), we haveAs , the right side of (30) tends to 0, and the left side of (30) also tends to 0, that is,

Lemma 3. It holds that

Proof. Now, for , we define two convex cones and bywhere , is a parameter, is the positive first eigenvalue of the operator , and is the imbedding constant for to :

Lemma 4. There exists such that if , then

Proof. Given , define byFirst, we prove . For , , then the problemthen, and in . Substitute into equation (37), we havefor all , . Hence, . Consequently, .
Next, we prove . The function satisfiesSubstituting by , thenthen, for all , , it holdsIt follows that ; therefore, . We have . Take as a test function in (37), thenBy and , we may assume that for some , thenNow, we choose , such thatThen, if and , we have . By equation (44), weWith equation (43) and all the above estimates, we have for ,Notice that , we obtainHence, for . Similarly, .

4. Sign-Changing Solutions

As we mentioned in the introduction, the operator will be used to construct descending flow for the functional . Since is merely continuous, we need the following modification of .

Lemma 5. Let be the critical point set of :Then, there exists a locally Lipschitz continuous operator defined on which inherits the operators of . More precisely,

Proof. The proof is similar to Lemma 1 of [15], we omit it.
By equations (39), (43) in Lemma 3, and (50), for some constants , one has

Lemma 6. (1)Assume sufficiently small. Then, for and , one has(2)Let be an arbitrary finite-dimensional subspace of , then for , one obtains

Proof. (1)By the conditions and , there exist constants such that for , one has(2)By the condition , there exists such that . Then, for , we haveas , where are constants.
Denote

Lemma 7. Let be an open neighborhood of . Then, there exists such that for , there exists a continuous map satisfying(1) for .(2) for .(3).(4)If the function is odd in , then for all .(5), for .

Proof. The proof is similar to the proof of Lemma 7 of [16], we omit it.

Proof of Theorem 1. Let be an open neighborhood of . Let , where is the map obtained in Lemma 7. Then, for some , the map satisfies.(1).(2).(3), .(4)If is even, then .That is, I has a deformation property. Moreover, by Lemma 6, I satisfies the symmetric mountain pass geometry. Now, Theorem 1 follows from Theorems 1 and 2 in [15].

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Acknowledgments

This work was supported by the Youth Natural Science Foundation of Shanxi Province (No. 20210302124527), the Science and Technology Innovation Project of Shanxi (No. 2020L0260), and Youth Science Foundation of Shanxi University of Finance and Economics (No. QN-202020).