© G. Zhang and S. Liu. 2008
Received: 31 October 2007
Accepted: 4 March 2008
Published: 12 March 2008
and for some . They obtained the decay, symmetry, and existence of solutions for problem (1.2). In 2004, Li and Yang  proved that problem (1.2) possesses at least a positive solution when the nonlinearities and are "asymptotically linear" at infinity and "superlinear" at zero, that is,
where is critical Sobolev exponent, and with for a domain containing the origin Here, denotes the open ball centered at the origin of radius . The existence of a nontrivial solution was obtained by using a generalized linking theorem.
As it is well known in dimensions the nonlinearities are required to have polynomial growth at infinity, so that one can define associated functionals in Sobolev spaces. Coming to dimension much faster growth is allowed for the nonlinearity. In fact, the Trudinger-Moser estimates in replace the Sobolev embedding theorem used in
As the potential and the nonlinearity are asymptotic to a constant function, Cao  obtained the existence of a nontrivial solution. As the potential and the nonlinearity are asymptotically periodic at infinity, Alves et al.  proved the existence of at least one positive weak solution.
Our aim in this paper is to establish the existence of a nontrivial solution for problem (P) in subcritical case. To our knowledge, there are no results in the literature establishing the existence of solutions to these problems in the whole space. However, it contains a basic difficulty. Namely, the energy functional associated with problem (P) has strong indefinite quadratic part, so there is not any more mountain pass structure but linking one. Therefore, the proofs of our main results cannot rely on classical min-max results. Combining a generalized linking theorem introduced by Kryszewski and Szulkin  and Trudinger-Moser inequality, we prove an existence result for problem (P).
The paper is organized as follows. In Section 2, we recall some results and state our main results. In Section 3, main result is proved.
2. Preliminaries and Main Results
Here, we assume the following condition:
Assume (H1), (H2), and (H3), and suppose
In the whole space do Ó and Souto  proved a version of Trudinger-Moser inequality, that is,
3. Proof of Theorem 2.2
Proof of Theorem 2.2.
This work is supported by Innovation Program of Shanghai Municipal Education Commission under Grant no. 08 YZ93.
- Colin F, Frigon M: Systems of coupled Poisson equations with critical growth in unbounded domains. Nonlinear Differential Equations and Applications 2006,13(3):369-384. 10.1007/s00030-006-4012-1MathSciNetView ArticleMATHGoogle Scholar
- Ding Y, Li S: Existence of entire solutions for some elliptic systems. Bulletin of the Australian Mathematical Society 1994,50(3):501-519. 10.1017/S0004972700013605MathSciNetView ArticleMATHGoogle Scholar
- de Figueiredo DG: Nonlinear elliptic systems. Anais da Academia Brasileira de Ciências 2000,72(4):453-469. 10.1590/S0001-37652000000400002MathSciNetView ArticleGoogle Scholar
- de Figueiredo DG, do Ó JM, Ruf B: Critical and subcritical elliptic systems in dimension two. Indiana University Mathematics Journal 2004,53(4):1037-1054. 10.1512/iumj.2004.53.2402MathSciNetView ArticleMATHGoogle Scholar
- de Figueiredo DG, Yang J: Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Analysis: Theory, Methods & Applications 1998,33(3):211-234. 10.1016/S0362-546X(97)00548-8MathSciNetView ArticleMATHGoogle Scholar
- Li G, Yang J: Asymptotically linear elliptic systems. Communications in Partial Differential Equations 2004,29(5-6):925-954.MathSciNetView ArticleMATHGoogle Scholar
- Adimurthi , Yadava SL:Multiplicity results for semilinear elliptic equations in a bounded domain of involving critical exponents. Annali della Scuola Normale Superiore di Pisa 1990,17(4):481-504.MathSciNetMATHGoogle Scholar
- de Figueiredo DG, Miyagaki OH, Ruf B:Elliptic equations in with nonlinearities in the critical growth range. Calculus of Variations and Partial Differential Equations 1995,3(2):139-153. 10.1007/BF01205003MathSciNetView ArticleMATHGoogle Scholar
- Shen Y, Yao Y, Chen Z:On a class of nonlinear elliptic problem with critical potential in . Science in China Series A 2004, 34: 610-624.MathSciNetGoogle Scholar
- Cao DM:Nontrivial solution of semilinear elliptic equation with critical exponent in . Communications in Partial Differential Equations 1992,17(3-4):407-435. 10.1080/03605309208820848MathSciNetView ArticleMATHGoogle Scholar
- Alves CO, do Ó JM, Miyagaki OH:On nonlinear perturbations of a periodic elliptic problem in involving critical growth. Nonlinear Analysis: Theory, Methods & Applications 2004,56(5):781-791. 10.1016/j.na.2003.06.003MathSciNetView ArticleMATHGoogle Scholar
- Kryszewski W, Szulkin A: Generalized linking theorem with an application to a semilinear Schrödinger equation. Advances in Differential Equations 1998,3(3):441-472.MathSciNetMATHGoogle Scholar
- Willem M: Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications. Volume 24. Birkhäuser, Boston, Mass, USA; 1996:x+162.Google Scholar
- Li G, Szulkin A: An asymptotically periodic Schrödinger equation with indefinite linear part. Communications in Contemporary Mathematics 2002,4(4):763-776. 10.1142/S0219199702000853MathSciNetView ArticleMATHGoogle Scholar
- do Ó JM, Souto MAS:On a class of nonlinear Schrödinger equations in involving critical growth. Journal of Differential Equations 2001,174(2):289-311. 10.1006/jdeq.2000.3946MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.