In this paper, we consider the Schrödinger-Kirchhoff-type problem
(1.1)
where constants , , or 3, and . We are concerned with the existence of nontrivial solutions of (1.1), corresponding to the critical points of the energy functional
(1.2)
where .
When , , problem (1.1) reduces to the following semilinear Schrödinger equation:
(1.3)
Equation (1.3) has been studied extensively by many authors, and there is a large body of literature on the existence and multiplicity of results of solutions for equation (1.3); for example, we refer the readers to [1–3] and references therein.
On the other hand, the Kirchhoff-type problem on a bounded domain ,
(1.4)
is related to the stationary analogue of the Kirchhoff equation
which was proposed by Kirchhoff [4] as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings. It is pointed out in [5] that Kirchhoff-type problem (1.4) models several physical and biological systems, where u describes the process which depends on the average of itself (for example, population density). For the case of bounded domain, some interesting studies by variational methods can be found in [6–14] for equation (1.4) with several growth conditions on f. Very recently, Kirchhoff-type equations on the unbounded domain or the whole space have also attracted a lot of attention. Many solvability conditions on the nonlinearity have been given to obtain the existence and multiplicity of solutions for problem (1.1). In [15, 16], the authors studied the case of superlinear nonlinearity. In [17, 18], the authors considered the case of radial potentials. In [19], the authors studied the case of nonhomogeneous nonlinearity.
Equation (1.1) can be viewed as the combination of (1.3) and (1.4) in . So we call it the Schrödinger-Kirchhoff-type problem. Compared with (1.3) or (1.4), problem (1.1) is much more complicated and is in place of the bounded domain in (1.4). This makes the study of equation (1.1) more difficult and interesting. In the present paper, the goal is to study the existence of nontrivial solutions for equation (1.1) for the case of asymptotical nonlinearity and weaker superlinear conditions compared with [15, 16].
For the potential V, we assume
-
(V)
satisfies and for each , , where α is a constant and meas denotes the Lebesgue measure in .
Set
with the norm
Denote
with the inner product and the norm
Since , it is easy to see that the Hilbert space E is continuously embedded in . By the Sobolev embedding theorem, we know the embedding
where , if , and , if , is also continuous, and there is a constant , , such that
(1.5)
where
Moreover, the embedding is compact for each due to the assumption (V). In fact, if , it follows from Lemma 3.4 in [20]. If , we also claim that the compactness of the embedding is valid for . Indeed, let be a sequence of E such that weakly in E. Similarly to the proof of Lemma 3.4 in [20], up to a subsequence, we can obtain strongly in . Next, we shall prove strongly in for . In fact, , there are and such that . Then, by the Hölder inequality,
Since the embedding
is continuous, is bounded in . This, together with , shows strongly in , , . Therefore, the compactness result holds for .
Throughout this paper, we shall always assume , . To establish the existence of nontrivial solutions for Schrödinger-Kirchhoff-type problem (1.1) in , we make the following assumptions:
(f1) and for some , where c is a positive constant.
(f2) as uniformly in .
(f3) uniformly in .
(f4) for all , where ( appears in (1.5)).
We consider the subcritical case in the present paper, (f2) and (f3) with characterize the asymptotic behavior of f at zero and infinity. The condition (f3) with implies , that is, 4-superlinearity of F at infinity.
As usual, the Ambrosetti-Rabinowitz [21] type condition
is assumed to ensure the boundedness of a Palais-Smale sequence. It implies that there exists a constant such that . By a simple calculation, it is easy to see that (AR) implies that , and hence (f3) with does not ensure the condition (AR). Furthermore, our condition (f3) with is much weaker than the condition (AR). It is important to show that there are many functions satisfying the conditions (f1)-(f4) with or , where
(1.6)
But the condition (AR) is not satisfied.
Example 1 For any given , set
Then it is easy to verify that satisfies (f1)-(f4) with , but does not satisfy the condition (AR).
Example 2 Set
Then it is easy to verify that satisfies (f1)-(f4) with , but does not satisfy the condition (AR).
Our main results are stated as the following theorems.
Theorem 1.1 Let conditions (V) and (f1)-(f4) hold and . Then problem (1.1) has at least one nontrivial solution in E.
Theorem 1.2 Let conditions (V), (f1)-(f4) with hold. Then problem (1.1) has at least one nontrivial solution in E.
Corollary 1.3 If the following (f5) or (f6) is used in place of (f4):
(f5) , , .
(f6) is nondecreasing with respect to t.
Then the conclusions of Theorem 1.1 and Theorem 1.2 hold.