Research Article | Open Access
Multiplicity of Solutions for Schrödinger Equations with Concave-Convex Nonlinearities
We study the multiplicity of solutions for a class of semilinear Schrödinger equations: where satisfies some kind of coercive condition and involves concave-convex nonlinearities with indefinite signs. Our theorems contain some new nonlinearities.
1. Introduction and Main Results
In this paper, we consider the multiplicity of solutions for the following semilinear Schrödinger equations:Equation (1) has many applications in mathematical physics. For instance, in finding the standing wave solutions for the following nonlinear Schrödinger equationwe can see that a standing wave solution of (2) is a solution of the form where . The function solves (2) if and only if solves (1) with and .
The existence and multiplicity of solutions for problem (1) have been studied by many mathematicians in last two decades [1–36]. In 1992, Coti Zelati and Rabinowitz  obtained the existence of infinitely many solutions for problem (1) when and are both periodic in and is supposed to satisfy the following so-called Ambrosetti-Rabinowitz superlinear condition.(AR)there exists such that Condition (AR) provided a global growth condition of at both origin and infinity, which plays an important role in showing the boundedness of Palais-Smale sequences and the geometrical structure for the corresponding functional. But (AR) is so strict that many functions do not satisfy this condition. An usual and weaker superlinear condition is(SQ) as uniformly in ,which is first introduced by Liu and Wang  to obtain multiple solutions for superlinear elliptic equations and has been used by many mathematicians. Via a Nehari-type argument, Li et al.  obtained a ground state solution for problem (1) with the help of the following Nehari type assumption:(Ne) is strictly increasing on and .
In a recent paper, (Ne) is weakened by Liu  when the author treated a class of periodic Schrödinger equations. He made the following assumption:(WN) is increasing on and .
After then, there are some papers [28, 35, 36] that obtained existence and multiplicity of nontrivial solutions for problem (1) with condition (WN). Recently, Tang  introduced a new superlinear condition.(Ta)there exists a such that
With (Ta), Tang obtained the existence of ground state solutions for a class of superlinear Schrödinger equation involving some new nonlinearities. Motivated by the above works, in this paper, we shall study the multiplicity of solutions of problem (1) with concave-convex nonlinearities and the superlinear term satisfies some different growth assumptions from above. There are only few papers considering the concave-convex nonlinearities for problem (1). In , Wu considered problem (1) in a bounded domain with concave-convex nonlinearities and obtained two positive solutions when the weight function is indefinite in sign. After then, Wu  considered problem (1) in the entire space with sign-changing weight and obtained multiple positive solutions for problem (1). The results on multiple solutions for problem (1) with concave-convex nonlinearities can be also found in [10, 13]. But in [10, 13, 30, 31], the authors only considered the specific nonlinearities. In this paper, we consider a more general case. The potential satisfies the following coercive condition which is introduced by Bartsch and Wang in :(), . There exists such that
The main purpose of this paper is to obtain multiplicity of solutions for problem (1) with some new nonlinearities. The nonlinear term is considered to satisfy the following form:Let and . Now we state our main results.
Theorem 1. Suppose that , (7), and the following conditions hold:() and .()There exit , , and such that for all .()For any , there exist such thatwhere and for .(), where and .()There exits such that in with .Then, there exists such that for any , problem (1) possesses at least two solutions.
Remark 2. Since , we can see that , which implies that , where for .
Remark 3. It is easy to see that does not satisfy (AR), (SQ), (WN), and (Ta) since can change sign.
Remark 4. In 2005, Liu and Wang  also considered problem (1) with concave-convex nonlinearities. But in their theorems, the nonlinear term was assumed to be a specific form, which is different from our theorem. Furthermore, it was required that in , which is not needed in .
Remark 6. It is easy to see that and are both indefinite in signs. The sign-changing nonlinear terms have been studied by Tang . But in , the author only considered the case and is positive when is large enough which is different from .
Theorem 7. Suppose that , (7), , , , , and the following conditions hold:(), for all .() as uniformly in .()There exist and such that ()There exist and such that () as uniformly in .Then, there exists such that for any , problem (1) possesses at least two solutions.
Remark 8. There are functions satisfying the conditions of , but not the condition . For example, let
Remark 11. Obviously, it can be, respectively, deduced from (AR), (WN), and (Ta) that(WSQ) for all However, (WSQ) cannot be deduced from the conditions of our theorems and there are functions to show this difference. For example, let andIt is easy to see that (12) satisfies the conditions , but not (WSQ).
In this paper, we will use the variational methods to prove our theorems. First, we introduce the definition of the condition and condition.
Definition 12. Let be a Hilbert space. A functional is said to satisfy the condition with respect to , , if any sequence satisfying implies a convergent subsequence, where is a sequence of linear subspace of with finite dimensional.
Definition 13. Let be a Hilbert space. A functional is said to satisfy the condition if for any sequence satisfying which is bounded and as possesses a convergent subsequence.
In our proof, the Mountain Pass Theorem and the following critical points theorems are employed.
Lemma 14 (Lu ). Let be a real reflexive Banach space and be a closed bounded convex subset of . Suppose that is a weakly lower semicontinuous (w.l.s.c. for short) functional. If there exists a point such that then there must be a such that
Lemma 15 (Chang ). Suppose that is even with and that()there are constants and a finite dimensional linear subspace such that ,()there is a sequence of linear subspace , , and there exists such that If, further, satisfies the condition with respect to , then possesses infinitely many distinct critical points corresponding to positive critical values.
In this paper, we let with the inner product and the norm . Then, is a Hilbert space. For any , we denote The embedding theorem shows that continuously for , which implies that there exists a constant such thatfor all . The corresponding functional is defined on asWith condition , we have the following compact embedding theorem.
Lemma 16 (see ). Under assumption , the embedding from into is compact for .
Lemma 17. Suppose that , , , and hold; then, the functional is well defined and of class withfor all , where and . Moreover, the critical points of in are solutions for problem (1).
Proof. By , , and , we havefor all . It follows from and (20) that there exists such thatThen, we can deduce that which implies that is well defined. Similar to the proof of Proposition in , we can see that and is compact. Obviously, is also of class and is compact, which means is of class and (22) holds. Finally, since is continuously embedded into , a standard argument shows that all critical points of on are solutions of (1). We finish the proof of this lemma.
3. Proof of Theorem 1
Subsequently, we show possesses the conditions of the Mountain Pass Theorem.
Lemma 19. Suppose the conditions of Theorem 1 hold; then, there exist such that for all , where .
Proof. By Lusin’s Theorem and , there exists such that is continuous in with with . We choose . Then, by (21), (25), , and , for any , we obtain where , which implies that Therefore, there exists such that . Let , we can see , which proves this lemma.
Lemma 21. Suppose the conditions of Theorem 1 hold; then, satisfies the condition.
Proof. Assume that is a sequence such that is bounded and as . Then, there exists a constant such thatSubsequently, we show that is bounded in . Arguing in an indirect way, we assume that as . It follows from (31), (27), (21), (23), , and that there exist such that which is a contradiction. Hence, is bounded in . Then, there exists a subsequence, still denoted by , such that in . Therefore, Let and satisfying , where . By , we can see that for . It follows from (20) and Lemma 16 that Similarly, we have It follows from (27) that which implies that as . Then, satisfies the condition.
Lemma 22. Suppose that the conditions of Theorem 1 hold; then, there exists a critical point of corresponding to negative critical value.
Proof. By Lemma 19, we can see that there exists a local minimizer of in , the following proof is to show this minimizer is not zero. By and , there exists such thatfor all and , where . Choosing , it follows from (21), (37), and that for small enough. By Lemmas 19 and 14, there exists such that The proof of this lemma is finished.
4. Proof of Theorem 5
Lemma 23. Suppose the conditions of Theorem 5 hold; then, I satisfies .
Proof. Let be a completely orthogonal basis of and , where . For any , we setIt follows from Lemma in  that as for any . SetThen, there exists such that for all . Then, for any with , it follows from (21), , (23), , and (40) thatHence, (42) shows that there exist and such that . We finish the proof of this lemma.
Lemma 24. Suppose the conditions of Theorem 5 hold; then, I satisfies .
Proof. Let and be as defined in Lemma 20. Then, it is easy to see that and is a Hilbert space. We can choose a sequence completely orthogonal basis . Let and . Then, for any , we have , where . Since , there exists a constant such thatfor all . We can deduce from (21), (25), , and (43) that Then, there exists such that for all , which proves this lemma.
The proof of the following lemma is similar to Lemma 21; we omit it here.
Lemma 25. Suppose the conditions of Theorem 5 hold; then, satisfies the condition.
5. Proof of Theorem 7
Lemma 26. Suppose the conditions of Theorem 7 hold; then, there exist such that for all .
Proof. Set such that , where is defined in Lemma 22. For , it follows from that there exist such that for all and . It follows from and that there exists such thatfor all and . It follows from and (48) that there exists such thatfor all . Then, we can deduce from (47) and (49) thatfor all . By (21), (50), (20), and (25), for every , we have which implies that Therefore, there exists such that and . Let , we can see , which proves this lemma.
Lemma 28. Suppose the conditions of Theorem 7 hold; then, satisfies the condition.
Proof. Assume that is a sequence such that Then, there exists a constant such thatSubsequently, we show that is bounded in . Set where is defined in . Arguing in an indirect way, we assume that as . Set ; then, , which implies that there exists a subsequence of , still denoted by , such that in and uniformly on as . The following discussion is divided into two cases.
Case 1 (). Let . Then, we can see that . Since as and ; then, we have as for a.e. . On one hand, it follows from (21), (25), and (54) that which implies thatOn the other hand, by , (49), and Fatou’s Lemma, we can obtain which contradicts (57).
Case 2 (). By , we can deduce that for all , which implies thatfor all . It follows from (54), (21), (25), , , (20), (60), and Sobolev’s embedding theorem that