Existence of positive ground states for some nonlinear Schrödinger systems
© Zhang et al.; licensee Springer. 2013
Received: 26 March 2012
Accepted: 12 January 2013
Published: 28 January 2013
We prove the existence of positive ground states for the nonlinear Schrödinger system
where a, b are periodic or asymptotically periodic and F satisfies some superlinear conditions in . The proof is based on the method of Nehari manifold and the concentration-compactness principle.
MSC:35J05, 35J50, 35J61.
1 Introduction and statement of the main result
Following the work  by Lin and Wei about the existence of ground states for the problem (1.2), there are many results on the existence of ground states relevant to five parameters (, , , and β); see [3–9] and the references therein. Later in , assuming , Pomponio and Secchi established the existence of radially symmetric ground states for (1.2) with general nonlinearities ( and ).
Moreover, in what follows, the notation inf (sup) is understood as the essential infimum (supremum). In the sequel, let and with , we always assume that
(V1) , ,
(F1) for some and ,
(F2) as ,
(F3) , , is strictly increasing,
(F4) as ,
(F5) , , , , ,
(F6) , .
(F1)-(F4) are similar to the conditions of the nonlinearities for the periodic system (1.3) as considered in . We divide the study of (NLS) into two cases as follows.
First, we consider the periodic case
(V2) , , , .
We have the following result.
Theorem 1.1 Let (V1), (V2) and (F1)-(F6) hold. Then the system (NLS) has a positive ground state.
Next, we consider the asymptotically periodic case. We assume that there are functions satisfying (V1) and (V2) and a, b satisfies that
(V3) , ,
(V4) , .
We have the following result.
Theorem 1.2 Assume that and satisfy (V2). Let (V1), (V3), (V4) and (F1)-(F6) hold. Then the system (NLS) has a positive ground state.
Remark 1.2 Conditions (V1) and (V4) imply that and satisfy (V1).
In addition, we consider the following conditions:
(V5) , ,
(F7) , , .
We have the following result.
Theorem 1.3 Suppose that and satisfy (V1) and (V2). Let (V1), (V3), (V5) and (F1)-(F7) hold. Then the system (NLS) has a positive ground state.
We will prove Theorems 1.1, 1.2 and 1.3 using the method of Nehari manifold. We first reduce the problem of seeking for ground states of (NLS) into that of looking for minimizers of the functional constrained on the Nehari manifold. Then we apply the concentration-compactness principle to solve the minimization problem. Since the Nehari manifold for (NLS) may not be smooth, in the same way as , we will make use of the differential structure of a unit sphere in to find a sequence (c is the infimum of the functional constrained on the Nehari manifold). When (NLS) is periodic, we will use the invariance of the functional under translation to recover the compactness of the sequence. When the system (NLS) is asymptotically periodic, the difficulty is to recover the compactness for the sequence. By comparing c with the infimum of the functional of the related periodic limit system constrained on the corresponding Nehari manifold, we will restore the compactness.
The paper is organized as follows. In Section 2 we give some preliminaries. In Section 3 we introduce the variational setting. In Section 4 we consider the periodic case and prove Theorem 1.1. Section 5 is devoted to studying the asymptotically periodic case and showing Theorems 1.2 and 1.3.
2 Notation and preliminaries
We use the following notation:
For simplicity, we denote and , where is measurable.
X denotes the Sobolev space (), with the standard scalar product and the norm . with the norm . When there is no possible misunderstanding, the subscripts could be omitted.
The usual norm in () will be denoted by .
For any and , denotes the ball of radius ϱ centered at z.
A ground state such that , (, ) is called a positive (non-negative) ground state. Below we give some lemmas useful for studying our problem.
Moreover, (F3) implies the function is increasing in for all .
Lemma 2.2 Let (F1) and (F2) hold. Then is weakly sequentially continuous. Namely, if in H, then in H.
Hence, is bounded in H. Combining with the fact that is dense in H, we easily deduce that (2.4) holds for any . Therefore, in H. □
3 Variational setting
This section is devoted to describing the variational framework for the study of ground states for (NLS).
Below we investigate the main properties of Φ on M.
Lemma 3.1 Let (F2) and (F3) hold. Then Φ is bounded from below on M by 0.
By (2.3) we have . □
Define the least energy of (NLS) on M by , then . Next, we prove M is a manifold. First, we give the following two lemmas, which will be important when proving M is a manifold.
since is bounded in H. □
for each , there exists such that if , then for and for ;
there exists such that for all ;
for each compact subset , there exists a constant such that for all .
- (i)Note that
- (ii)If , then
We argue by contradiction. Suppose that there exist a compact set W and a sequence such that and . Since W is compact, there exists such that in H. Then Lemma 3.2 implies that . Contrary to Lemma 3.1 since . This ends the proof. □
As a consequence of Lemma 3.3(i), we can define the mapping by . By Lemma 3.3, [, Proposition 3.1(b)] yields the following result.
Lemma 3.4 If (V1) and (F1)-(F4) are satisfied, then m is a homeomorphism between S and M, and M is a manifold.
If M is a manifold, we can make use of the differential structure of M to reduce the problem of finding a ground state for (NLS) into that of looking for a minimizer of and solve the minimizing problem. However, since , M may not be a manifold. Noting that M and S are homeomorphic, we will take advantage of the differential structure of S to seek for ground states for (NLS) as . Therefore, as in , we introduce the functional defined by , and we have the following conclusion.
If is a PS sequence for Ψ, then is a PS sequence for Φ.
is a critical point of Ψ if and only if is a nontrivial critical point of Φ. Moreover, .
A minimizer of is a solution of (NLS).
Proof As in the proof of [, Corollary 3.3], we can show (i) and (ii). Now, we prove the conclusion (iii). Indeed, let such that . Then , where . By the conclusion (ii), we have . So, . Using the conclusion (ii) again, we deduce that . □
From the definition of a ground state, we translate the problem of looking for a ground state for (NLS) into that of seeking for a solution for (NLS) which is a minimizer of . By Proposition 3.1(iii), in order to look for a ground state for (NLS), we just need to seek for a minimizer of .
4 The periodic case
In this section, we consider the periodic case and prove Theorem 1.1. In , Szulkin and Weth considered the existence of ground states for periodic single Schrödinger equations. Treating as in , we find ground states for a periodic case for the system (NLS). In addition, under conditions (F5) and (F6), we deduce that there are positive ground states.
From the statement in Section 3, it suffices to solve the minimizing problem. By conclusions (i) and (ii) of Proposition 3.1, we first make use of the minimizing sequence of Ψ to obtain a sequence of Φ. Then we use the invariant of the functional under translation of the form , to recover the compactness for the sequence.
Proof of Theorem 1.1 Let be a minimizing sequence of Ψ. By the Ekeland variational principle [, Theorem 8.5], we may assume that . Using Proposition 3.1(i), we have that , where . Proposition 3.1(ii) implies that .
- (i)If , then for any , there exists such that , for . Combining with (2.2), for and , we have
- (ii)If , then we can assume that in . From the Lions compactness lemma [, Lemma 1.21], it follows that there exist and such that(4.1)
Since Φ and M are invariant by translation of the form , , translating if necessary, we may assume is bounded. Since in , then (4.1) implies . Then from Lemma 3.2, we deduce that . This is impossible since .
Hence, is bounded in H. Suppose that in H, in and a.e. on for a subsequence. Since , Lemma 2.2 yields .
where (4.3) follows from the Fatou lemma and (2.3). Then . According to , we have . Thus, . Consequently, is a ground state of (NLS).
It remains to look for a positive ground state for (NLS). First, we can assume that is non-negative. In fact, note that and for all . Then . Let be such that . By (F6) we easily have that . Moreover, since . Then . So, is also a minimizer of Φ on M. Then is also a ground state of (NLS). Thus we can assume that is a non-negative ground state for (NLS). Now, we claim that , . Indeed, if , then from (F5) and , the first equation of (NLS) yields that . Then . This is impossible. So, . Similarly, . By (F5), applying the maximum principle to each equation of (NLS), we infer that , . The proof is complete. □
5 The asymptotically periodic case
As for c, we have .
Lemma 5.1 Suppose that , satisfy (V1) and (V2). Let (F1)-(F6) hold. Then the problem (NLS) p has a positive ground state such that .
Proof As a corollary of Theorem 1.1, we infer that the problem (NLS) p has a positive ground state. Moreover, from the argument of Theorem 1.1, we find that the ground state of the problem (NLS) p we obtained is a minimizer of on . □
Next, we prove that under some conditions.
Lemma 5.2 Suppose that , satisfy (V2). Let (V1), (V4), (5.2) and (F1)-(F6) hold. Then .
If , we are done. Otherwise, . Then by (5.3) and (5.4), we get and . Then is a ground state for (NLS). Note that is a solution of (NLS) p . From the first equations of (NLS) and (NLS) p , we infer that . Similarly, contrary to (5.2). The proof is now complete. □
Lemma 5.3 Suppose that , satisfy (V1) and (V2). Let (V1), (V5), (5.2) and (F1)-(F7) hold. Then .
Then . Below we argue analogously with the proof of Lemma 5.2 to infer that . This ends the proof. □
Now, we are ready to prove Theorems 1.2 and 1.3. The proof is partially inspired by , where the authors dealt with Schrödinger-Poisson equations.
Proof of Theorem 1.2 As the argument of Theorem 1.1, we infer that there exists a sequence such that and .
This is a contradiction.
Hence, is bounded in H. Up to a subsequence, we assume that in H, in and a.e. on . By Lemma 2.2, we have . Namely, is a solution of (NLS).
Let be such that . We claim that for large n and .
which is impossible. Consequently, (5.7) holds.
Hence, . Then using (5.6), we have . Then . However, Lemma 5.2 implies that . This is a contradiction. Note that this contradiction follows from the hypothesis that . So, . Then .
where the inequality (5.13) holds by (2.3) and the Fatou lemma. Then . According to , we have . Then is a ground state for (NLS). Below we argue analogously with the proof of Theorem 1.1 to get a positive ground state for (NLS). The proof is complete. □
Proof of Theorem 1.3 By Lemma 5.3, repeating the argument of Theorem 1.2, we show the existence of a ground state for (NLS) and then look for a positive ground state as the argument of Theorem 1.1. □
The authors would like to express their sincere gratitude to the referee for helpful and insightful comments. Hui Zhang was supported by the Research and Innovation Project for College Graduates of Jiangsu Province with contract number CXLX12_0069, Junxiang Xu and Fubao Zhang were supported by the National Natural Science Foundation of China with contract number 11071038.
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