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Existence of infinitely many solutions for generalized Schrödinger-Poisson system
Boundary Value Problems volume 2014, Article number: 196 (2014)
We study the nonlinear generalized Schrödinger-Poisson system: , in , , in , where and are non-negative functions. The function is superlinear. Under appropriate assumptions on , , and , we prove the existence and multiplicity of nontrivial solutions by the variant fountain theorem established by Zou. Some recent results due to different authors are extended.
MSC: 35J20, 35J65, 35J60.
Consider the existence and multiplicity of nontrivial solutions for the nonlinear generalized Schrödinger-Poisson system:
where . Setting , (1.1) represents the well-known Schrödinger-Poisson system:
Such a system arises in an interesting physical context. If we look for solitary solutions of the Schrödinger equation for a particle in an electrostatic field, we just need to solve (1.2). We refer the interested readers to ,  for more details of the physical aspects.
With the aid of variational methods, under various hypotheses on , , and , system (1.2) has been extensively investigated over the past several decades. See for example, Benci and Fortunato , D’Aprile and Mugnai , Ambrosetti and Ruiz , Ambrosetti , and the references therein.
In recent years, there has been a lot of research on the existence of solutions for system (1.2) with and the potential being radially symmetric or nonradial. In , , the authors proved the existence of infinitely many pairs of high energy radial solutions when , and also obtained some existence results for . Sun  studied the existence of infinitely many solutions when . The authors of  proved the existence of positive solutions without compactness conditions if . Azzollini and Pomponio  proved the existence of a ground state solution to (1.2) for .
Furthermore, Sun et al. studied system (1.2) for being asymptotically linear and obtained ground state solutions. In , Huang et al. obtained the existence of at least a pair of fixed sign solutions and a pair of sign-changing solutions to system (1.2) involving a critical nonlinearity. Ding et al. studied (1.2) with a nonhomogeneous term, where is either asymptotically linear or asymptotically 3-linear with respect to u at infinity. Very recently, Liu and Guo  studied (1.2) with critical growth and obtained the existence of ground state solutions via variational methods.
The problem of finding infinitely many large energy solutions is a very classical problem. There is an extensive literature concerning the existence of infinitely many large energy solutions of (1.2). Chen and Tang  obtained infinitely many large energy solutions by the following variant ‘Ambrosetti-Rabinowitz’ type condition (AR for short)
where . After that, in , the authors studied (1.2) without the (AR) condition by the variant fountain theorem established by Zou , Theorem 2.1]. Later, Li and Chen in  also obtained infinitely many large energy solutions of (1.2) with without the (AR) condition.
By using the method of a cut-off function and the variational arguments, the authors  studied the following Schrödinger-Poisson system:
where is a bounded domain with smooth boundary ∂ Ω, , , is a continuous function and . They proved the existence and multiplicity results assuming on f a subcritical growth condition and also they considered the existence and nonexistence results in the critical case. Lately, Li and Zhang  discussed (1.1) with and being constants and obtained the existence of a positive radially symmetric solution without compactness conditions.
Motivated by the above facts, in the present paper our aim is to study the existence of infinitely many solutions for system (1.1). To the best of our knowledge, the existence and multiplicity of nontrivial solutions to system (1.1) has never been studied by variational methods, where g is a more general function, is also general, and and may be non-radial symmetrical and non-periodic. Before we state our main result, we list some conditions as follows, which have a role to play.
satisfies , where is a constant. Moreover, for any , , where denotes the Lebesgue measure in .
, and for any .
(g1) and there exist constants , such that
(g3) for all , and there exists a constant such that
(f1) and there exist constants , such that
where is the critical exponent for the Sobolev embedding in dimension 3.
(f2) uniformly for .
(f3), and uniformly for .
(f4) For a.e. , there exists a constant such that
Our main result reads as follows.
Assume that (V), (K), (g1)-(g3), and (f1)-(f5) hold; the problem (1.1) possesses infinitely many nontrivial solutionssatisfying
It is well known that the (AR) condition is used to guarantee the boundedness of (P.S.) sequences of the corresponding functional. However, there are functions satisfying the assumptions (f1)-(f5), but not satisfying the (AR) condition, for instance, , where .
Li et al. used
(f′4):, , .
to solve the problem (1.2). We claim that our condition (f4) is more general than (f′4). In fact, setting , we find that is increasing in with respect to t. Moreover, the function satisfies (f4) but not satisfies (f′4), where .
It is easy to present many functions satisfying (g1)-(g3), for example, , , , and so on. Moreover, setting , in (1.1), we can obtain similar results to the problem (1.1) in . But our proof is different from . For this reason, we use a small step.
Since we have the lack of the (AR) condition, in order to obtain the boundedness of (P.S.) sequences (see the proof of Theorem 1.1), we assume the range of α in (g1) is , i.e. α is subquadratic, but that of reference  is superquadratic and there exist some functions which satisfy the condition (f) in  that do not satisfy the conditions (g1) and (g3), for example, .
The outline of the paper is as follows: in Section 2, we present some preliminary results, which are necessary for Section 3. In Section 3, we give the proof of Theorem 1.1. Throughout the paper we shall denote by various positive constants.
In this section we outline the variational framework for the problem (1.1) and give some preliminary lemmas. Define the function space
Then E is a Hilbert space equipped with the inner product and norm
Since is bounded from below, the E is continuously embedded into for all . Therefore, there exists a positive constant such that
is the norm of the usual Lebesgue space . Moreover, by (V), the embedding is also compact for any , Lemma 3.4]. Let be the completion of with respect to the norm
It is well known that the embedding is continuous (see ).
It is clear that system (1.1) is the Euler-Lagrange equations of the functional defined by
Obviously, the action functional J belongs to and its critical points are the solutions of (1.1); see for instance . For any , by the Lax-Milgram theorem, we can obtain the result that the second equation in (1.1) has a unique solution . Substituting to the first equation of the problem (1.1), then the problem can be transformed to a one variable equation. In fact, we firstly get the following lemma.
For any, we have
By the condition (g1), we find that there exists such that
Then, by the Minkowski inequality and (2.1), we have
For any , the linear functional is defined as
By the Sobolev embedding theorem, , , and (2.3), we have
So, is continuous on . Hence, the Lax-Milgram theorem implies that, for every , there exists a unique such that
Using integration by parts, we get
in a weak sense. We can write an integral expression for in the form
The proof is complete. □
We consider the functional defined by . By (2.5), the reduced functional takes the form
By (f1) and (f2), for any , there exists such that, for all , ,
Therefore, by (f3), we obtain
Then, by Lemma 2.1, I is well defined and is a functional with derivative given by
Now, we can apply Theorem 2.3 of  to our functional I and obtain the following.
The following statements are equivalent:
is a solution of (1.1);
u is a critical point of I and .
Since we do not assume the (AR) condition, the verification of the (P.S.) condition becomes complicated, so we use the following variant fountain theorem introduced in  without the (P.S.) condition to handle the problem (1.1).
Let E be a Banach space withandwithfor any. Set, and.
Consider the following functional defined by
where are two functionals. Suppose that
(F1) maps bounded sets to bounded sets uniformly for . Furthermore,
(F2) for all , and or as .
(F3) There exist such that
where . Moreover, for a.e. , there exists a sequence such that
3 Proof of Theorem 1.1
In order to apply Theorem 2.1 to prove our main result, we define the functional on our working space E by
for all and . Then for all , as .
We choose a completely orthonormal basis of E and let for all . Then , can be defined as those in Section 2. Note that , where I is the functional defined in (2.6). We further need the following lemmas.
Let (V), (K), (g3), and (f1) be satisfied, then there exist a positive integerand a sequenceassuch that
Since E is compactly embedded into and , we have (see , Lemma 3.8])
It follows from (3.4) that there exists a positive integer such that , . Then we have
For each , choose
The proof is complete. □
Under the assumptions of (V), (K) and (f1)-(f3), then for the positive integerand the sequenceobtained in Lemma 3.1, there existsfor eachsuch that
It follows from (f3) that, for any , there exists such that, for all , ,
From (f1) and (f2), there exists such that, for all and ,
Then, by the mean value theorem, for all , , we obtain
For , by Lemma 2.1 and (3.12), we have
where in the last inequality we use the equivalence of all norms on the finite dimensional subspace . Let us choose M large enough such that . Then, when M is fixed, is also fixed. Since , we can choose such that
The proof is complete. □
Proof of Theorem 1.1
It follows from Lemma 2.1 and (2.8) that maps bounded sets to bounded sets uniformly for . By (g2) and (f5), for all . Thus, it follows from Lemmas 3.1 and 3.2 that the conditions of Theorem 2.1 are satisfied. Hence, for a.e. , there exists a sequence such that
with and .
Furthermore, it follows from the proof of Lemma 3.1 that
In view of (3.14), we can choose with and obtain the corresponding sequences (denoted by ) satisfying
Claim 1. The sequence has a strong convergent subsequence.
Claim 2. The sequence is bounded.
If not, without loss of generality, we suppose that . Let , then, up to a sequence, in view of the compact embedding of E into , , we have
Case 1. in E. As in , we choose such that
For any , we set . By (2.8) and (f3), we have
Then, choosing n sufficiently large such that
Thus, . In view of the choice of , we know that . Then, by (g3) and (f4), we have
where . This is a contradiction according to (3.19).
as , a contradiction to (3.19) again. Then is bounded in E.
In view of Claim 2 and (3.19), using similar arguments to the proof of Claim 1, we can also show that the sequence has a strong convergent subsequence with the limit being just a critical point of . Obviously, . Since as , we know that is an unbound sequence of critical points of functional I. Thus, the proof of Theorem 1.1 is complete. □
Benci V, Fortunato D: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 1998, 11: 283-293.
D’Aprile T, Mugnai D: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc. R. Soc. Edinb., Sect. A 2004, 134: 893-906. 10.1017/S030821050000353X
Ambrosetti A, Ruiz D: Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. 2008, 10: 391-404. 10.1142/S021919970800282X
Ambrosetti A: On Schrödinger-Poisson systems. Milan J. Math. 2008, 76: 257-274. 10.1007/s00032-008-0094-z
Sun J: Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 2012, 390: 514-522. 10.1016/j.jmaa.2012.01.057
Cerami G, Vaira G: Positive solutions for some non-autonomous Schrödinger-Poisson systems. J. Differ. Equ. 2010, 248: 521-543. 10.1016/j.jde.2009.06.017
Azzollini A, Pomponio A: Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J. Math. Anal. Appl. 2008, 345: 90-108. 10.1016/j.jmaa.2008.03.057
Sun J, Chen H, Nieto J: On ground state solutions for some non-autonomous Schrödinger-Poisson systems. J. Differ. Equ. 2012, 252: 3365-3380. 10.1016/j.jde.2011.12.007
Huang L, Rocha EM, Chen J: Positive and sign-changing solutions of a Schrödinger-Poisson system involving a critical nonlinearity. J. Math. Anal. Appl. 2013, 408: 55-69. 10.1016/j.jmaa.2013.05.071
Ding L, Li L, Zhang J:Multiple solutions for nonhomogeneous Schrödinger-Poisson systems with asymptotical nonlinearity in . Taiwan. J. Math. 2013, 17: 1627-1650. 10.11650/tjm.17.2013.2798
Liu Z, Guo S: On ground state solutions for the Schrödinger-Poisson equations with critical growth. J. Math. Anal. Appl. 2014.
Chen S, Tang C: High energy solutions for the Schrödinger-Maxwell equations. Nonlinear Anal. 2009, 71: 4927-4934. 10.1016/j.na.2009.03.050
Li Q, Su H, Wei Z: Existence of infinitely many large solutions for the nonlinear Schrödinger-Maxwell equations. Nonlinear Anal. 2010, 72: 4264-4270. 10.1016/j.na.2010.02.002
Zou W: Variant fountain theorems and their applications. Manuscr. Math. 2001, 104: 343-358. 10.1007/s002290170032
Li L, Chen S: Infinitely many large energy solutions of superlinear Schrödinger-Maxwell equations. Electron. J. Differ. Equ. 2012, 224: 1-9. 10.1016/j.jde.2005.07.012
Azzollini, A, d’Avenia, P, Luisi, V: Generalized Schrödinger-Poisson type system. , [arXiv:1009.2728v3]
Li F, Zhang Q: Existence of positive solutions to the Schrödinger-Poisson system without compactness conditions. J. Math. Anal. Appl. 2013, 401: 754-762. 10.1016/j.jmaa.2013.01.002
Zou W, Schechter M: Critical Point Theory and Its Applications. Springer, New York; 2006.
Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.
Evans LC: Partial Differential Equations. Am. Math. Soc., Providence; 1998.
Benmlih K:Stationary solutions for a Schrödinger-Poisson system in . Proceedings of the 2002 Fez Conference on Partial Differential Equations 2002, 65-76.
Benci V, Fortunato D, Masiello A, Pisani L: Solitons and the electromagnetic field. Math. Z. 1999, 232: 73-102. 10.1007/PL00004759
Jeanjean L:On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on . Proc. R. Soc. Edinb., Sect. A 1999, 129: 787-809. 10.1017/S0308210500013147
This article was supported by Natural Science Foundation of China 11271372 and by Hunan Provincial Natural Science Foundation of China 12JJ2004.
The authors declare to have no competing interests.
All authors, LX and HC, contributed to each part of this work equally and read and approved the final version of the manuscript.
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Xu, L., Chen, H. Existence of infinitely many solutions for generalized Schrödinger-Poisson system. Bound Value Probl 2014, 196 (2014). https://doi.org/10.1186/s13661-014-0196-1
- generalized Schrödinger-Poisson system
- variant fountain theorem
- variational approaches