Research Article | Open Access
Multiplicity of Solutions for a Modified Schrödinger-Kirchhoff-Type Equation in
We study the existence of infinitely many solutions for a class of modified Schrödinger-Kirchhoff-type equations by the dual method and the nonsmooth critical point theory.
1. Introduction and Main Result
In this paper, we study the following modified Schrödinger-Kirchhoff-type equations of the formwhere , , , , and .
When , (1) is reduced to the following Kirchhoff-type problem:If , problem (2) is related to the stationary analogue of the Kirchhoff equationproposed by Kirchhoff in  as an existence of the classical D’Alembert’s wave equation for free vibrations of elastic string. Kirchhoff’s model takes into account the changes for free vibrations of elastic strings. Recently, there have been many papers concerned with the Kirchhoff-type problems by variational methods; see [2–8] and the references therein. Many studies of them are concentrated on a bounded smooth domain of ; it is well known that the embedding is not compact. Hence, if we look for solution by variational methods, it is very difficult to prove condition. Moreover, in order to check the condition or some of its variants, one has to impose certain conditions.
When and , (1) is reduced to the following modified nonlinear Schrödinger equation:Solutions of (4) are standing waves of the following quasilinear Schrödinger equations: where is a real constant and and are real functions. Equations (5) are derived as model of several physical phenomena, such as [9, 10]. Many achievements had been obtained on the existence of ground states, infinitely many solutions, and soliton solutions for (4), by a dual approach, Nehari method, and the minimax methods in critical point theory, applying the perturbation approach and the Lusternik-Schnirelmann category theory; see [11–18]. Recently, K. Wu and X. Wu  obtained infinitely many small energy solutions of (1) by applying Clark’s theorem to a perturbation functional. In (1), set , in which and are real parameters, ; Liang and Shi  obtained soliton solutions of (1) by using the concentration-compactness principle and minimax methods.
In this paper, we transform (1) to another equation with a continuous energy functional in some Banach space. We obtain the existence of multiple solution for problem (1) via using nonsmooth critical point theory and using some new techniques. Throughout this paper, the main ideas used here come from Colin and Jeanjean  and Liu et al. .
We need the following several notations. Let with the inner productand the normLetand the norm
Let the following assumption hold:, , and .
Moreover, we need the following assumptions:()Let ; is odd in .()There exist constants and such thatfor all , where is the Sobolev critical exponent.() uniformly in as .()There exists such that , and , where .
The main result of this paper is as follows.
Theorem 1. Assume that and are satisfied. Then problem (1) has a sequence of solutions such that as .
Throughout the paper, and denote the strong and weak convergence, respectively. , , , and express distinct constants. For , the usual Lebesgue space is endowed with the norm
The paper is organized as follows. In Section 2, we reformulate our problem into a new one which has an associated functional well defined in a suitable space and present some preliminary results. In Section 3, we introduce some notions and results of nonsmooth critical point theory and we show that the functional satisfies condition. In Section 4, we complete the proof of Theorem 1.
2. The Dual Variational Framework and Preliminary Results
The energy functional corresponding to problem (1) is defined as follows: It should be pointed out that the main difficulty in treating this class of quasilinear equations in is the lack of compactness and the second-order nonhomogeneous term which prevents us from working directly in a classical function space. is not defined in ; thus we may not apply directly the variational method to study (1). To move these obstacles, we make the change of variable (see [12, 14]); that is, we consider , where is defined byIn order to prove our main result, we need some further properties of the function .
Lemma 2. The function has the following properties:(1) is uniquely defined, , and invertible;(2) for all ;(3) for all ;(4); ;(5) for all ;(6), for all ; for all ;(7)there exists a positive constant such that (8) for all ;(9)there exist positive constants , such that (10) is convex and for all and ;(11), , s. t. ;(12) is decreasing for .
After this change of variables, the functional is well defined onwhich is a Banach space endowed with the norm
Proposition 3. (1) There exists a positive such that, for all , (2) If in , then(3) If a.e. and then(4) If in , then in , for
(5) Under assumption , the embedding is compact for , and the embedding is continuous for .
(6) The embedding is continuous. Moreover, is dense in .
Lemma 4. Assume that conditions , , and hold; then, one has for the following assertions:(1) is continuous.(2)For every and , the derivation of in the direction at exists and will be denoted by(3)The map satisfies the following:(i) is linear in ;(ii) is continuous in ; that is, if in , then as .
Remark 5. Consider the following:
Proof. Let be a sequence such that in . By Proposition 3, we have Moreover, in for , in , and, up to subsequence, , a.e. .
, by and , there exists such thatThus, by Lemma 2 and (27), By Lemma of  and Lebesgue’s theorem, one hasBy Lemma 2 and Lebesgue’s theorem,thusConsequently, and is continuous in . Similarly, it can be proved that is continuous in for each . The proof of (24) is standard by using conditions , , and . It is obvious that is linear in . The proof is completed.
3. Nonsmooth Critical Framework
In the following, will denote a metric space endowed with the metric .
Definition 6. Let be a continuous function and let . One denotes by the supremum of the ’s in such that there exist and a continuous mapsatisfyingfor all , where is the open ball of center and of radius . The extended real number is called the weak slope of at .
If is a Finsler manifold of class and , it turns out that .
The function is lower semicontinuous.
Definition 7. Let be a continuous function. A point is called a critical point of if . A real number is called a critical value of if there exists a such that and .
Definition 8. Let be a continuous function and let . One says that satisfies the condition if each sequence with and has a convergent subsequence.
We can state a generalized version of the symmetric Mountain Pass Theorem for the case of continuous functionals.
Theorem 9 (see ). Let be an infinite-dimensional Banach space and let be continuous, even, and satisfying condition for every . Assume the following:(i)There exist , , and a subspace of finite codimension such that (ii)For every finite-dimensional subspace , there exists such that Then there exists a sequence of critical values of with as .
Now consider the functional given in the previous section.
Lemma 10. Consider , .
Proof. We use a similar argument in the proof of Theorem in . Let andIf , the conclusion is obvious. Otherwise, take such that . Then there exists a point with and . By Lemma 4, there exist such that for every . Let and . It is obvious that is continuous,for . Hence, by Definition 6, and by the arbitrariness of . The proof is completed.
Lemma 11. Under assumptions , , and , if satisfies , , then .
Proof. By (25), we havefor all .
For any , let if and if . For any , take in the above equality; then it can be deduced from (38) that By Lemma 2 and the Sobolev embedding theorem, where is positive constant.
For any , by (27), , Lemma 2, Hölder inequality, and Sobolev inequality, for simplicity, taking , we have where and are positive constants.
Set . Because , . Moreover, by (39), (40), and (41), we have Taking the limit in (42), Set and , . Then as and Becauseis convergent as , let ; then as . Hence Let ; then, we have where the positive constant is independent of . The proof is completed.
Corollary 12. If is a critical point of , then .
Lemma 13. , under assumptions , , and , the functional satisfies condition.
Let be a sequence with and . Then, it follows from Lemma 10 thatMoreover,Note thatLetting andwe deduce that for some positive constant . From (49) and (50), we havefor each . Since , by and Lemma 2, , we have where . Their estimates imply that is bounded in .
Hence, up to a subsequence, in . Because , from (49) and (50), Because for each , set ; it follows from Lemma 11 that . Consequently, By Proposition 3, we get in for . Using (27), Lemma 2, and Lebesgue’s theorem, we obtainBy assumption , Lemma 2, the Fatou Lemma, and the Lebesgue dominated convergence theorem, one hasThese imply thatNotice that in ; then, we conclude that in . By Proposition 3 and (59), we have We obtain in ; that is, the functional satisfies condition.
4. Proof of the Main Result
The following lemma implies that possesses the symmetric Mountain Pass Geometry.
Lemma 14. Suppose that , , and are satisfied. Then the functional satisfies condition (i) of Theorem 9.
Proof. Let , . If , by Lemma 2, we have which implies thatfor . Note that if with . Hence By , (27), Lemma 2, and Proposition 3, let ; then, we have for and . Hence condition (i) in Theorem 9 holds for small .
Lemma 15. Assume that , , and hold; then the functional satisfies condition (ii) of Theorem 9.
Proof. By Lemma 2, for , such that