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Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory
Boundary Value Problems volume 2014, Article number: 212 (2014)
In this paper, we study the following fourth-order elliptic equations of Kirchhoff type: , in , , where are constants, we have the potential and the nonlinearity . Under certain assumptions on and , we show the existence and multiplicity of negative energy solutions for the above system based on the genus properties in critical point theory.
MSC: 35J20, 35J65, 35J60.
1 Introduction and main results
Consider the following fourth-order elliptic equations of Kirchhoff type:
where a, b are positive constants. We assume that the functions , , and its primitive satisfy the following hypotheses.
(f1) , and there exist and positive functions , such that
(f2) There exist a nonzero measure open set and three constants , and such that
(f3) , .
(f4) There exist a nonzero measure open set and three constants and such that
Let , replace by a bounded smooth domain and set on Ω, then problem (1.1) is reduced to the following fourth-order elliptic equations of Kirchhoff type:
which is related to the following stationary analogue of the equation of Kirchhoff type:
where is the biharmonic operator. In one and two dimensions, (1.3) is used to describe some phenomena appearing in different physical, engineering and other sciences because it is regarded as a good approximation for describing nonlinear vibrations of beams or plates (see , ). By using the fixed point theorems in cones of ordered Banach spaces, Ma  considered the existence and multiplicity of positive solutions for the fourth-order equation:
Recently, by the variational methods, Ma and Wang etc. studied (1.4) and the following fourth-order equation of Kirchhoff type:
and obtained the existence and multiplicity of solutions; see –. Very recently, Wang et al. considered the existence of nontrivial solutions of (1.2) with one parameter λ in  by using the mountain pass techniques and the truncation method.
We note that problem (1.1) with , , and being replaced by , reduces the well-known fourth-order elliptic equations
There are some results for (1.5). For example, see –. By the mountain pass theorem and symmetric mountain pass theorem, Yin and Wu  obtained infinitely many high energy solutions for problem (1.5) under the condition that is superlinear at infinity in u. In order to overcome lack of compactness for the Sobolev’s embedding theorem in the whole space case, they assumed that the potential satisfies
(V0): such that and for any , .
Later, under the condition (V0), Ye and Tang  obtained the existence of infinitely many large-energy and small-energy solutions, which unifies and improves the results in . They also considered the sublinear case. Very recently, Zhang et al.  established the existence of infinitely many solutions by using the genus properties. The solvability of (1.1) without has also been well studied by various authors (see  and the references therein).
Motivated by the above works described, the object of this paper is to study the existence and multiplicity solutions for a class of sublinear fourth-order elliptic equation of Kirchhoff type by using the genus properties in critical theory. Our spirit is similar to , . Our main results are the following.
Assume that (V), and (f1)-(f2) hold, then the problem (1.1) possesses at least a nontrivial solution.
Assume that (V), and (f1)-(f3) hold, then the problem (1.1) possesses infinitely many negative energy solutions.
Obviously, we see that (f4) implies (f2). Then we have the following corollary.
Assume that (V), (f1), and (f4) hold, then the problem (1.1) possesses at least a nontrivial solution. If additionally (f3) holds, then the problem (1.1) possesses infinitely many negative nontrivial solutions.
It is well known that assumption (V0) implies a coercive condition on the potential , which was firstly introduced in  and is used to overcome the lack of compactness of embedding of the working space. In other words, under the weaker condition (V), the Sobolev embedding is not compact, which is a difficulty we must overcome.
The conditions (V) and (f1)-(f4) were introduced in ,  to obtain the existence of infinitely many solutions for fourth-order elliptic equations and sublinear Schrödinger-Maxwell equations. An interesting question now is whether the same existence results occur to the nonlocal problem (1.1). We now give a positive answer. Moreover, let , and be replaced by in problem (1.1); we will get the main results in .
According to Theorem 1.2, the nonlinearity can be allowed to indefinite sign-changing. For example, let and , , , where . Obviously,
where , then (f1), (f2) and (f3) are satisfied by choosing
To the best of our knowledge, little has been done for the existence of infinitely many nontrivial solutions to problem (1.1) by using the genus properties in critical theory.
The outline of the paper is given as follows: in Section 2, we present some preliminary results. The proofs of our main results are given in Section 3. Throughout this paper, C denotes various positive constants.
with the inner product and the norm
Define our working space
with the inner product and norm
where is an equivalent to the norm . Since the embedding () is continuous, there exists such that , .
We define the functional
where . Then we have the following lemma.
Under the conditions (V) and (f1), I is of classand
for all. Furthermore, ifis a critical point of the functional I, thenis a solution of the problem (1.1).
By (f1), one gets
Thus, I is well defined on E. We now prove that (2.2) holds. For any and the function , it follows from (f1) and the Hölder inequality that
This implies that (2.2) holds. Next, we prove that is continuous. Let in E. By the embedding theorem, one has
Firstly, we show that
Otherwise, there exist and a subsequence such that
In fact, by (2.5), passing to a subsequence if necessary, it can be assumed that . Set , , then . Obviously,
It follows from (2.8), (2.9), and the Lebesgue dominated convergence theorem that (2.6) holds. Secondly, set with the norm . Then the embedding is continuous. By the continuity of the embedding and the boundedness of , one has
as , which implies the continuity of . Furthermore, by standard arguments, we can prove that is a solution of (1.1) if and only if u is a critical point of the functional I. The proof is complete. □
In order to deduce our results, we need to quote a few results.
Let E be a real Banach space andsatisfy the-condition. If I is bounded from below, thenis a critical value of I.
Let E be a Banach space, and. Set
For , we say genus of A is n (denoted by ) if there is an odd map and n is the smallest integer with this property.
Let I be an evenfunctional on E and satisfy the-condition. For any, set
If and , then is a critical value of I.
If there exists such that and , then .
3 Proofs of main results
According to Theorem 2.1, we need the following lemma.
Assume that (V) and (f1)-(f2) hold, then I is bounded from below and satisfies the-condition.
By (2.1), (f1), the Sobolev embedding theorem, and the Hölder inequality, one has
Since , (3.1) suggests as . Thus, I is bounded from below. We now prove that I satisfies the -condition. Assume that is a sequence such that is bounded and as . It follows from (3.1) that there exists a constant such that
So it can be assumed that
For any given number , by (f1), let us choose such that
For another, we obtain from (f1), (3.2), the Sobolev embedding theorem and the Hölder inequality that
By (2.2), we get
Proof of Theorem 1.1
By Lemmas 2.1 and 3.1, the conditions of Theorem 2.1 are satisfied. Thus, is a critical value of I, that is, there exists a critical point such that . Now, we show . Choose , then by (2.1) and (f2), for , we have
Since , it follows from (3.10) that for small enough. Hence , which implies being a nontrivial critical point of I with . That is to say, that is a nontrivial solution of (1.1). The proof is completed. □
Proof of Theorem 1.2
By Lemma 3.1, is bounded from below and satisfies the -condition. It follows from (f3) that I is even and . In order to apply Theorem 2.2, we now show that for any there exists ε such that
For any , we take n disjoint open sets such that . For each , choose such that and
Then, for each , there exist , , such that
Then we get
Since all norms are equivalent in a finite dimensional normed space, there exists a positive constant such that
Since , it follows from (3.16) that there exist and such that
By (3.17), we know that for , which, together with the fact that and is even, implies that
Thus, (3.11) holds. Let . By (3.18) and I being bounded from below on E, then , that is to say, for any , is a real negative number. It follows from Theorem 2.2 that I has infinitely many nontrivial critical points. Thus, problem (1.1) possesses infinitely many nontrivial negative energy solutions. □
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This article was supported by Natural Science Foundation of China 11271372 and by Hunan Provincial Natural Science Foundation of China 12JJ2004.
The authors declare to have no competing interests.
All authors, LX and HC, contributed to each part of this work equally and read and approved the final version of the manuscript.
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Xu, L., Chen, H. Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory. Bound Value Probl 2014, 212 (2014). https://doi.org/10.1186/s13661-014-0212-5
- fourth-order elliptic equations of Kirchhoff type
- genus theory
- variational methods