Research Article | Open Access
Existence of Nontrivial Solutions for Periodic Schrödinger Equations with New Nonlinearities
We study the Schrödinger equation: , where is -periodic and is -periodic in the -variables; is in a gap of the spectrum of the operator . We prove that, under some new assumptions for , this equation has a nontrivial solution. Our assumptions for the nonlinearity are very weak and greatly different from the known assumptions in the literature.
1. Introduction and Statement of Results
In this paper, we consider the following Schrödinger equation: where . For and , we assume the following. (v) is 1-periodic in for , 0 is in a spectral gap of , and and lie in the essential spectrum of . Denote () is 1-periodic in for . And there exist constants and such that where () The limit holds uniformly for . And there exists such that where , . () For any , , where () There exist and such that, for every with , and, for every with ,
Remark 1. By the definitions of and , it is easy to verify that, for all , Together with as and , this implies that
Remark 2. There are many functions satisfying . We give several examples here.
Example 1. , , , and
Example 2. , , , and
Example 3. , , and .
A solution of (1) is called nontrivial if . Our main results are as follows.
Theorem 3. Suppose and are satisfied. Then (1) has a nontrivial solution.
Note that the limits and hold uniformly for .
Implying , we have the following corollary.
Corollary 4. Suppose , , , , and are satisfied. Then (1) has a nontrivial solution.
It is easy to verify that the condition , for every . And the assumption that as uniformly for imply and . Therefore, we have the following corollary.
Corollary 5. Suppose , , , and are satisfied. Then (1) has a nontrivial solution.
Semilinear Schrödinger equations with periodic coefficients have attracted much attention in recent years due to its numerous applications. One can see [1–24] and the references therein. In , the authors used the dual variational method to obtain a nontrivial solution of (1) with , where is an asymptotically periodic function. In , Troestler and Willem firstly obtained nontrivial solutions for (1) with being a function satisfying the Ambrosetti-Rabinowitz condition:(AR)there exists such that, for every , , where , , and with . Then, in , Kryszewski and Szulkin developed some infinite-dimensional linking theorems. Using these theorems, they improved Troestler and Willem’s results and obtained nontrivial solutions for (1) with only satisfying and the (AR) condition. These generalized linking theorems were also used by Li and Szulkin to obtain nontrivial solution for (1) under some asymptotically linear assumptions for (see ). In  (see also ), existence of nontrivial solutions for (1) under and the (AR) condition was also obtained by Pankov and Pflüger through approximating (1) by a sequence of equations defined in bounded domains. In the celebrated paper , Schechter and Zou combined a generalized linking theorem with the monotonicity methods of Jeanjean (see ). They obtained a nontrivial solution of (1) when exhibits the critical growth. A similar approach was applied by Szulkin and Zou to obtain homoclinic orbits of asymptotically linear Hamiltonian systems (see ). Moreover, in  (see also ), Ding and Lee obtained nontrivial solutions for (1) under some new superlinear assumptions on different from the classical (AR) conditions.
Our assumptions on are very weak and greatly different from the assumptions mentioned above. In fact, our assumptions do not involve the properties of at infinity. It may be asymptotically linear growth at infinity, that is, , or superlinear growth at infinity as well, that is, . Moreover, the assumptions allow in a neighborhood of (see Remark 2).
In this paper, we use the generalized linking theorem for a class of parameter-dependent functionals (see [17, Theorem 2.1] or Proposition 8 in the present paper) to obtain a sequence of approximate solutions for (1). Then, we prove that these approximate solutions are bounded in and (see Lemmas 13 and 14). Finally, using the concentration-compactness principle, we obtain a nontrivial solution of (1).
Notation. denotes the open ball of radius and center . For a Banach space , we denote the dual space of by and denote strong and weak convergence in by and , respectively. For , we denote the Fréchet derivative of at by . The Gateaux derivative of is denoted by , . denotes the standard space , and denotes the standard Sobolev space with norm . We use , to mean , .
2. Existence of Approximate Solutions for (1)
Under the assumptions , , and , the functional is of class on , and the critical points of are weak solutions of (1).
Assume that holds, and let be the self-adjoint operator acting on with domain . By virtue of , we have the orthogonal decomposition such that is negative (resp., positive) in (resp., in ). As in [5, Section 2] (see also [6, Chapter 6.2]), let be equipped with the inner product and norm , where denotes the inner product of . From , with equivalent norms. Therefore, continuously embeds in for all if and for all if . In addition, we have the decomposition where is orthogonal with respect to both and . Therefore, for every , there is a unique decomposition with and Moreover, The functional defined by (14) can be rewritten as where
Let be the total orthonormal sequence in . Let , be the orthogonal projections. We define on . The topology generated by is denoted by , and all topological notation related to it will include this symbol.
Lemma 6. Suppose that holds. Then (a), where is defined in ;(b)for any , there exists with such that .
Proof. (a) We apply an indirect argument, and assume by contradiction that
From assumption , is in the essential spectrum of the operator (with domain ):
Then, by Weyl’s criterion (see, e.g., [25, Theorem VII.12] or [26, Theorem 7.2]), there exists a sequence with the properties that , and .
Since , we deduce that for all . Together with the facts that is a continuous periodic function and , this implies It follows that there exists a constant such that Note that Together with (29) and the fact that and , this implies . It contradicts , . Therefore, .
(b) It suffices to prove that From (21), we deduce that . From assumption , is in the essential spectrum of . By Weyl’s criterion, there exists such that and . Multiplying by and then integrating it into , by (20) and (22), we get that It follows that . Multiplying by and then integrating it into , we get that It implies that . This together with implies .
Let and From assumption (5), we have . Together with the result of Lemma 6, this implies that . Choose Then by the result of Lemma 6, there exists with such that Set Then, is a submanifold of with boundary
Definition 7. Let . A sequence is called a Palais-Smale sequence at level (-sequence for short) for , if and as .
Proposition 8. Let . The family of -functional has the form Assume(a), ;(b) as ;(c) for all , is -sequentially upper semicontinuous; that is, if , then and is weakly sequentially continuous. Moreover, maps bounded sets to bounded sets;(d) there exist with and such that, for all , Then there exists such that the Lebesgue measure of is zero and, for every , there exist and a bounded -sequence for , where satisfies
For and , let Then and it is easy to verify that a critical point of is a weak solution of where
Lemma 9. Suppose that and hold. Then, there exist and such that the Lebesgue measure of is zero and, for every , there exist and a bounded -sequence for , where satisfies
Proof. For , let
Then, and satisfy assumptions (a) and (b) in Proposition 8, and, by (43), .
From (43) and (20), for any and , we have Let and be such that . It follows that , , and . In addition, up to a subsequence, we can assume that a.e. in . Then, we have By Remark 1, for all and . This together with Fatou’s lemma implies Then, by (49), we obtain This implies that is -sequentially upper semicontinuous.
If in , then, for any fixed , as , This implies that is weakly sequentially continuous. Moreover, it is easy to see that maps bounded sets to bounded sets. Therefore, satisfies assumption (c) in Proposition 8.
Finally, we will verify assumption (d) in Proposition 8 for .
From assumption and as uniformly for , we deduce that, for any , there exists such that From (49) and (55), we have, for , Then by the Sobolev inequality and (by (21) and (22)), we deduce that there exists a constant such that Choose and such that and . Then, for every , we have Let be such that and . Then, from (58), we deduce that, for ,
We will prove that as and . Arguing indirectly, assume that, for some sequences and with , there is such that for all . Then, setting , we have , and, up to a subsequence, , and .
First, we consider the case . Dividing both sides of (49) by , we get that
From (5) and the result of Lemma 6, we deduce that where is defined by (34). Note that, for , we have . This implies that, when is large enough, By (10), we have, when is large enough, Combining the above two inequalities yields We used the inequalities in the second inequality of (64).
Since for some , by (36), we get that Note that, by the choice of (see (35)), we have . Then by (64) and the fact that , we have that It contradicts (60), since as .
Second, we consider the case . In this case, . It follows that since and . Therefore, the right hand side of (60) is less than when is large enough. However, as , the left hand side of (60) converges to zero. It induces a contradiction.
Therefore, there exists such that This implies that satisfies assumption (d) in Proposition 8 if . Finally, it is easy to see that Then, the results of this lemma follow immediately from Proposition 8.
Lemma 10. Suppose that and are satisfied. Let be fixed, where is the constant in Lemma 9. If is a bounded -sequence for with , then, for every , there exists such that, up to a subsequence, satisfies
Proof. The proof of this lemma is inspired by the proof of Lemma 3.7 in . Because is a bounded sequence in , up to a subsequence, either (a) or(b)there exist and such that .
If (a) occurs, using the Lions lemma (see, e.g., [21, Lemma 1.21]), a similar argument as for the proof of [19, Lemma 3.6] shows that It follows that On the other hand, as is a -sequence of , we have and . It follows that This contradicts (73). Therefore, case (a) cannot occur.
If case (b) occurs, let . For every , Because and are -periodic in every , is still bounded in , Up to a subsequence, we assume that in as . Since in , it follows from (75) that . Recall that is weakly sequentially continuous. Therefore, and, by (76), .
Finally, by and Fatou’s lemma
Lemma 11. There exist and such that, for any , if satisfies , then .
Proof. We adapt the arguments of Yang [23, p. 2626] and Liu [12, Lemma 2.2]. Note that, by and , for any , there exists such that Let be a critical point of . Then is a solution of Multiplying both sides of this equation by , respectively, and then integrating into , we get that It follows that where and are positive constants related to the Sobolev inequalities and . From the above two inequalities, we obtain Because , this implies that for some if and are small enough and . The desired result follows.
Lemma 12. Suppose and are satisfied. Then, there exist , , and such that ,
3. A Priori Bound of Approximate Solutions and Proof of the Main Theorem
Lemma 13. Suppose and are satisfied. Let be the sequence obtained in Lemma 12. Then, and
Proof. From , we deduce that is a weak solution of (45) with ; that is,
By assumption and the bootstrap argument of elliptic equations, we deduce that .
Multiplying both sides of (85) by and integrating into , we get that Recall that and . Then by (5), we get that This together with (86) yields . It follows that on .
Similarly, multiplying both sides of (85) by and integrating into , we can get that on . Therefore, for all .
Lemma 14. Suppose that , , , , and are satisfied. Let be the sequence obtained in Lemma 12. Then
Proof. As and