The Existence Of Multiple Solutions For Semilinear Schr(?)dinger Equation Without Ambrosetti-Rabinowitz Condition

In this paper, we study the existence of multiple solutions to the following semilinear Schrodinger equation: where V:RN→R is a bounded locally Holder continuous function satisfying V(x)≥a for some a>0,f∈C0(RN×R,R) is superlinear at t=0 and subcritical at t=∞.We prove that under suitable conditions the problem ((?)) has at least one non-trivial solution without assuming the Ambrosetti-Rabinowitz condition and ((?)) has infinitely many distinct solutions if, in addition,f is odd in t. Our result is stronger than a similar result of A. Szulkin and T. Weth in [26]. In the mean time, our main result extends partially a result about the nonlinear elliptic problem in a bounded domain by G. Li and C. Yang in [16] to the semilinear Schrodinger equation.