Existence of infinitely many solutions for generalized Schrödinger-Poisson system
© Xu and Chen; licensee Springer 2014
Received: 22 April 2014
Accepted: 31 July 2014
Published: 25 September 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.
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.
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.
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.
satisfies , where is a constant. Moreover, for any , , where denotes the Lebesgue measure in .
, and for any .
where is the critical exponent for the Sobolev embedding in dimension 3.
(f2) uniformly for .
(f3), and uniformly for .
Our main result reads as follows.
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.
It is well known that the embedding is continuous (see ).
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.
The proof is complete. □
Now, we can apply Theorem 2.3 of  to our functional I and obtain the following.
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.
where are two functionals. Suppose that
(F2) for all , and or as .
3 Proof of Theorem 1.1
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.
The proof is complete. □
The proof is complete. □
Proof of Theorem 1.1
with and .
Claim 1. The sequence has a strong convergent subsequence.
Claim 2. The sequence is bounded.
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. □
This article was supported by Natural Science Foundation of China 11271372 and by Hunan Provincial Natural Science Foundation of China 12JJ2004.
- Benci V, Fortunato D: An eigenvalue problem for the Schrödinger-Maxwell equations. Topol. Methods Nonlinear Anal. 1998, 11: 283-293.MathSciNetMATHGoogle Scholar
- 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/S030821050000353XMathSciNetView ArticleMATHGoogle Scholar
- Ambrosetti A, Ruiz D: Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math. 2008, 10: 391-404. 10.1142/S021919970800282XMathSciNetView ArticleMATHGoogle Scholar
- Ambrosetti A: On Schrödinger-Poisson systems. Milan J. Math. 2008, 76: 257-274. 10.1007/s00032-008-0094-zMathSciNetView ArticleMATHGoogle Scholar
- 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.2012.01.057MathSciNetView ArticleMATHGoogle Scholar
- Cerami G, Vaira G: Positive solutions for some non-autonomous Schrödinger-Poisson systems. J. Differ. Equ. 2010, 248: 521-543. 10.1016/j.jde.2009.06.017MathSciNetView ArticleMATHGoogle Scholar
- 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.2008.03.057MathSciNetView ArticleMATHGoogle Scholar
- 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.2011.12.007MathSciNetView ArticleMATHGoogle Scholar
- 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.2013.05.071MathSciNetView ArticleMATHGoogle Scholar
- 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.17.2013.2798MathSciNetView ArticleMATHGoogle Scholar
- Liu Z, Guo S: On ground state solutions for the Schrödinger-Poisson equations with critical growth. J. Math. Anal. Appl. 2014.Google Scholar
- Chen S, Tang C: High energy solutions for the Schrödinger-Maxwell equations. Nonlinear Anal. 2009, 71: 4927-4934. 10.1016/j.na.2009.03.050MathSciNetView ArticleMATHGoogle Scholar
- 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.2010.02.002MathSciNetView ArticleMATHGoogle Scholar
- Zou W: Variant fountain theorems and their applications. Manuscr. Math. 2001, 104: 343-358. 10.1007/s002290170032View ArticleMathSciNetMATHGoogle Scholar
- 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.2005.07.012View ArticleMATHGoogle Scholar
- 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.2013.01.002MathSciNetView ArticleMATHGoogle Scholar
- Zou W, Schechter M: Critical Point Theory and Its Applications. Springer, New York; 2006.MATHGoogle Scholar
- Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.View ArticleMATHGoogle Scholar
- Evans LC: Partial Differential Equations. Am. Math. Soc., Providence; 1998.MATHGoogle Scholar
- Benmlih K:Stationary solutions for a Schrödinger-Poisson system in . Proceedings of the 2002 Fez Conference on Partial Differential Equations 2002, 65-76.Google Scholar
- Benci V, Fortunato D, Masiello A, Pisani L: Solitons and the electromagnetic field. Math. Z. 1999, 232: 73-102. 10.1007/PL00004759MathSciNetView ArticleMATHGoogle Scholar
- 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/S0308210500013147MathSciNetView ArticleMATHGoogle Scholar