Nonlinear Analysis 86 (2013) 52–57 Contents lists available at SciVerse ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Generalized saddle point theorem and asymptotically linear problems with periodic potential ✩ Shibo Liu a,∗ , Zupei Shen b a School of Mathematical Sciences, Xiamen University, Xiamen 361005, China b School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China article info Article history: Received 19 June 2012 Accepted 11 March 2013 Communicated by S. Carl MSC: 35J60 35J20 58E05 Keywords: Schrödinger equations Spectral gap Generalized saddle point theorem (PS) c sequences abstract We prove a critical point theorem, which is an infinite dimensional generalization of the classical saddle point theorem of P. H. Rabinowitz. As an application we obtain solution of asymptotically linear Schrödinger equations with periodic potential. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction In this paper we consider the following stationary Schrödinger equation  −∆u + V (x)u = λu + g (x)f (u) + h(x), u ∈ H 1 (R N ). (1.1) We are interested in the asymptotically linear case lim |u|→∞ f (u) u = 0. (1.2) Then, it is well known that under suitable assumptions the functional Φ(u) = 1 2  R N  |∇u| 2 + V (x)u 2 + λu 2  dx −  R N g (x)F (u)dx −  R N h(x)udx (1.3) with F (u) =  u 0 f (τ)dτ being of class C 1 on X = H 1 (R N ), and critical points of Φ are weak solutions of (1.1). ✩ Supported by National Natural Science Foundation (11171204) of China, the Fundamental Research Funds for the Central Universities, and GDNSF (S2012010010038). ∗ Corresponding author. Tel.: +86 592 2580627; fax: +86 592 2580608. E-mail addresses: liusb@xmu.edu.cn, laosb@qq.com (S. Liu). 0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.03.005