Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory
© Xu and Chen; licensee Springer 2014
Received: 24 January 2014
Accepted: 3 September 2014
Published: 25 September 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
(f3) , .
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.
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 .
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.
where is an equivalent to the norm . Since the embedding () is continuous, there exists such that , .
where . Then we have the following lemma.
for all. Furthermore, ifis a critical point of the functional I, thenis a solution of the problem (1.1).
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.
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.
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.
Proof of Theorem 1.1
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
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. □
This article was supported by Natural Science Foundation of China 11271372 and by Hunan Provincial Natural Science Foundation of China 12JJ2004.
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