Nontrivial Solutions for Periodic Schrödinger Equations with Sign-Changing Nonlinearities
We prove a new infinite-dimensional linking theorem. Using this theorem, we obtain nontrivial solutions for strongly indefinite periodic Schrödinger equations with sign-changing nonlinearities.
1. Introduction and Statement of Results
In this paper, we consider the following semilinear Schrödinger equation: where . The potential is 1-periodic in for . In this case, the spectrum of the operator is a purely continuous spectrum bounded from below and consists of closed disjoint intervals ([1, Theorem XIII.100]). Thus, the complement consists of open intervals called spectral gaps. More precisely, for , 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 For , we assume the following.(f1) is a Caratheodory function in and it is 1-periodic in for . In addition, there exist constants and such that where (f2)The limits and hold uniformly for .
Let , , , and (f3)There exist and such that
A solution of (1) is called nontrivial if . Our main result is the following theorem.
Theorem 1. Suppose that (v) and (f1)–(f3) are satisfied. Then problem (1) has a nontrivial solution.
As an application of this theorem, we have the following corollary.
Corollary 2. Suppose that and (v) is satisfied. Then, there exists such that, for , has a nontrivial solution.
Remark 3. This corollary shows that, under assumptions (f1)–(f3), the nonlinearity may be sign-changing.
Under assumption (v), the quadratic form has infinite-dimensional negative and positive spaces. This case is called strongly indefinite. Semilinear periodic Schrödinger equations with strongly indefinite linear parts have attracted considerable attention in recent years due to their numerous applications in mathematical physics. In , the authors used a dual variational method to obtain a nontrivial solution of (1) with , where is a perturbed periodic function and . In , Troestler and Willem used critical point theory to obtain a nontrivial solution of (1) by assuming that , for some and satisfy the so-called Ambrosetti-Rabinowitz condition , , where . Kryszewski and Szulkin  subsequently proved a new infinite-dimensional linking theorem. Using this, they generalized Troestler and Willem’s result by assuming that and satisfy the Ambrosetti-Rabinowitz condition. Similar results were obtained by Pankov and Pflüger in [5, 6] by using an approximation method and a variant of the Nehari method. Equation (1), with asymptotically linear nonlinearities and other superlinear nonlinearities, has also been studied by several researchers. The interested readers can see [7–10] for the asymptotically linear case and [9, 11–17] for the superlinear case. Moreover, (1), with belonging to the spectrum of , was investigated in [8, 18, 19]. It is worth mentioning that methods of studying strongly indefinite periodic Schrödinger equations can shed light on other strongly indefinite problems, such as Hamiltonian systems and discrete nonlinear Schrödinger equations. The reader can consult [8, 20] or  for more details.
All existence results for (1) mentioned above are obtained under the assumption that does not change sign in ; that is, in or in . However, under our assumptions (f1)–(f3), can be negative in . Together with (7), this implies that may change sign in . As we know, this situation has never before been studied. And it is the novelty of our main results of Theorem 1 and Corollary 2. Theorem 1 can be seen as a generalization of Theorem 3 of .
The difficulties of (1) with regard to sign-changing nonlinearity are due to two aspects. The first is that the classical infinite-dimensional linking theorem (see [23, Theorem 6.10] or ) cannot be used to deal with (1) in this case. To use this linking theorem, the functional corresponding to (1) must satisfy a certain upper semicontinuous assumption. However, when is sign-changing, the functional corresponding to (1) does not satisfy such a assumption. The second aspect that causes problems is that sign-changing nonlinearity creates more difficulties in proving the boundedness of the Palais-Smale sequence.
In this paper, we provide a variant of the classical infinite-dimensional linking theorem (see Theorem 7). This theorem and its corollary replace the upper semicontinuous assumption in the classical infinite-dimensional linking theorem ([23, Theorem 6.10]) with other assumptions. We present the theorem and its proof in Section 2. Using this theorem, a sequence (see Definition 6) of (1) is obtained. Under (f1)–(f3), we can prove that this sequence is bounded in (see Lemma 12). We then obtain a nontrivial solution of (1) from the sequence by using the concentration-compactness principle.
Notation. denotes an 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 and .
2. A New Infinite-Dimensional Linking Theorem
Before stating this theorem, we give some notations and definitions.
Let be a separable Hilbert space with inner product and norm , respectively. are closed subspaces of and . Let be the total orthonormal sequence in . Let be the orthogonal projections. We define on . Then and if is bounded and , then weakly converges to in . The topology generated by is denoted by , and all topological notations related to it will include this symbol.
Let and with . Set Then, is a submanifold of with boundary
Definition 4. A functional is called -upper semicontinuous if, for any , is a -closed set. And is called weakly sequentially continuous if and are such that ; then, for any , .
Remark 5. If a functional has the form with , and is weakly sequentially lower semicontinuous, that is, if , then , then by Remark 2.1 (iii) of , is -upper semicontinuous.
Definition 6. Let . A sequence is called a sequence for , if
The main results of this section are the following theorem and corollary.
Theorem 7. If satisfies the following(a) is weakly sequentially continuous,(b)there exists a -upper semicontinuous functional such that , ,(c)there exist with and such that then, for any , there exists a sequence of with such that
Corollary 8. Let and satisfy assumptions (a), (b), and (c) in Theorem 7. If then there exist and a sequence of with such that
Remark 9. (i) To use the classical infinite-dimensional linking theorems, such as Theorem 3.4 of  or Theorem 6.10 of , to get a Palais-Smale sequence or sequence for a functional satisfying the linking condition (18), this functional should be -upper semicontinuous. This assumption precludes applying these classical infinite-dimensional linking theorems to problems possessing wider class of nonlinearities, such as nonlinear Schrödinger equations with sign-changing nonlinearities.
(ii) sequence (see Definition 6) can be seen as a weighted variation of Palais-Smale sequence. It plays important role in proving the boundedness. For example, the sequence in Lemma 11 of this paper cannot be replaced with Palais-Smale sequence.
(iii) This theorem and its corollary are new infinite-dimensional linking theorems different from the one published in [24, Theorem 1.3].
Proof of Theorem 7. Arguing indirectly, assume that the result does not hold. Then, there exist and such that
Since in , we deduce that Therefore,
Step 1. A vector field in a -neighborhood of
For every , there exists with such that . Then, (23) implies that From the definition of , we deduce that if a sequence -converges to , that is, , then in (see Remark 6.1 of ). By the weakly sequential continuity of , we get that for any , . This implies that is -sequentially continuous in . By (30), the -sequential continuity of in , and the weakly lower semicontinuity of the norm , we get that there exists a -open neighborhood of such that
Because is a bounded convex closed set in the Hilbert space , is a -closed set. It follows that is a -open set. Moreover, since is a -upper semicontinuous functional, is a -closed set. It follows that is a -open set. Therefore, the family is a -open covering of . Let Then, is a -open neighborhood of .
Since is metric, hence paracompact, there exists a local finite -open covering of finer than . If for some , we choose and if or if , we choose . Let be a -Lipschitz continuous partition of unity subordinated to . And let Since the -open covering of is local finite, each belongs to only finitely many sets . Therefore, for every , the sum in (35) is a finite sum. This implies the following.(a)For any , there exist a -open neighborhood of and such that is contained in a finite-dimensional subspace of and This means that is locally Lipschitz continuous and -locally Lipschitz continuous.Moreover, by the definition of and (31), we get the following.(b)For every , and for every ,
Step 2. From (36) and the fact that , , we have This implies that is a local Lipschitz mapping under the norm. Then, by the standard theory of ordinary differential equation in Banach space, we deduce that the following initial value problem has a unique solution in , denoted by , with the right maximal interval of existence . Furthermore, using (36) and the Gronwall inequality (see, e.g., Lemma 6.9 of ), the similar argument as the proof of (c) in [23, Lemma 6.8] yields that(A) is -continuous; that is, if , , , and satisfy and , then .
From , (see (37)), we have Therefore, is nonincreasing along the flow . It follows that if ; that is, for any and any , is still in . Moreover, since is a neighborhood of , for any and ; that is, the flow with cannot leave . Therefore, is an invariant set of the flow . Then, , (see (37)) and Theorem of  implies that, for any , .
Step 3. We will prove that where is defined in (28).
Let . By the result in Step 2, we have and Together with , (see (37)), this yields Then, by the Gronwall inequality (see, e.g., Lemma 6.9 of ), we get that Since , by (45) and the definition of (see (29)), we obtain (42).
Step 4. From (26) and (38), we deduce that
We show that, for any , . Arguing indirectly, assume that this was not true. Then, there exists such that . Since is nonincreasing along the flow , from (42), we deduce that . Then, by (46), This contradicts . Therefore, we have(B).Moreover, using the result (a) in Step 1 and the fact that is -continuous (see (A)), the similar argument as the proof of the result (b) of [23, Lemma 6.8] yields that(C)each point has a -neighborhood such that is contained in a finite-dimensional subspace of .
Step 5. Let where , and are defined in (11), (14), and (15). Then From (see (18)) and the fact that, for any , the function is nonincreasing, we deduce that , . Therefore, Since has properties (A) and (C) obtained in Steps 2 and 4, respectively, and satisfies (51), there is an appropriate degree theory for (see Proposition 6.4 and Theorem 6.6 of ). Then, the same argument as the proof of Theorem 6.10 of  yields that It follows that and . Therefore, there exists such that . It contradicts property (B) obtained in Step 4, since . This completes the proof of this theorem.
Proof of Corollary 8. Since , we can choose such that . By Theorem 7, there exists a sequence of such that and . From , we deduce that . Since is -upper semicontinuous, is a -open set. Therefore, there exists such that Since , by (53), we obtain .
3. Variational and Linking Structure for (1)
Under assumptions (v), (f1) and the first part of (f2), the functional belongs to class for . The derivative of is and the critical points of are weak solutions of (1).
Assume that (v) holds and let be the self-adjoint operator acting on with domain . By virtue of (v), we have the orthogonal decomposition such that is positive (resp., negative) in (resp., in ). Let be equipped with the inner product and norm , where denotes the inner product of . From (v), with equivalent norms. Therefore, continuously embeds into for all . In addition, we have the decomposition where and and are orthogonal with respect to both and . Let and be orthogonal projections. Therefore, for every , there is a unique decomposition with and Moreover, Therefore, We denote Let Then, by (54) and (61), can be written as The derivative of is given by
Let be the total orthonormal sequence in and let be the norm defined by (12). The topology generated by is denoted by , and all topological notations related to it will include this symbol.
Lemma 10. Suppose (v) and (f1)–(f3) are satisfied. Then(i)there exist , , and with such that where , and are defined in (14) and (15);(ii)there exists a -upper sem-continuous functional (see Definition 4) defined in such that and in .
Proof. We divide the proof into several steps.
Step 1. We will prove that there exists such that .
From (f1) and (f2), we deduce that, for any , there exists such that Then by the Sobolev inequality , , and the definition of (see (66)), there exists such that, for any , Choose in (70) and let . We then see that, for ,
Step 2. We will prove that Let and . From (7), we deduce in . Let From (9), we have Then, From (76), (63), (66), and the fact that in , we deduce that, for any ,
Now, we prove that as and . Together with (72), this yields that there exists such that .
Arguing indirectly, assume that, for some sequence with , there is such that for all . Then, setting , we have and, up to a subsequence, , and . Dividing both sides of by , we obtain By (76) and (63), we have Combining (78) and (79), we obtain
We first consider the case . From (see (f2)), we have Note that, for , we have . Together with (81), this implies that By , we obtain Combining (82) and (83) yields This contradicts (80), since as .
We then consider the case . In this case, . It follows that since and . Therefore, the right hand side of (80) is less than when is sufficiently large. However, as , the left side of (80) converges to zero. This also induces a contradiction.
Step 3. From (76) and (63), we deduce that, for any , Let Then by , and Remark 5, we deduce that is -upper semicontinuous. Moreover, , and, by (86), in .
Combining Steps 1–3, we obtain the desired results of this lemma.
4. Boundedness of Sequence and Proof of the Main Results
According to Definition 6, a sequence is called a sequence of if
Lemma 11. Suppose that (v) and (f1)–(f3) are satisfied. Let be a sequence of . Then where and and are from (f1) and (f3), respectively.
Proof. Let and . Then It follows that . Together with the fact that is a sequence for , this implies that By (6), Then, by (55) and (91) and the fact that on , we obtain where in . Together with the Sobolev inequality , this yields Because and on , we obtain from (94) the result Similarly, we can prove that where . The result of this lemma follows from (96) and (97).
Lemma 12. Suppose that (v) and (f1)–(f3) are satisfied. Let be a sequence of . Then
Proof. From , we get that . Then, by (67), we have
It follows that
From (f1) and (f2), we deduce that there exists such that
Note that and are orthogonal with respect to . Then, by (63), we have
and recall that (see (74)). Using the Hölder inequality, from (100), (101), and (102), we have
where the positive constant is from the Sobolev inequality , . By Lemma 11, we have
Combining (105) with (104) yields
From and , we obtain Together with (8), this implies