Existence Of Nontrivial Solutions For Some Nonlinear Schr(?)dinger Type Equations

The main research work of this paper is as follows:1. The following Schrodinger-Poisson system in R3 with potential indefinite in sign and a general 4-sublinear nonlinearity is studied. Its variational functional does not satisfy the Palais-Smale condition under our assumptions. We obtain a sequence of approximate solutions by using a finite-dimensional approximation method. Then we show that these approximate solutions converge weakly to a nontrivial solution of this system.2. The following p-Laplace equation in RN is studied, where N≥2 and 1<q<p<N. We assume the potential V2 is locally bounded function and satisfy that there exist positive constants L1, C1, C2 such that C1(1+|x|)-s<V2(x)≤C2(1+|x")-s for |x|>L1. We prove that the Palais-Smale condition holds for the functional corresponding to this equation. We obtain infinitely many solutions by using pseudo-index theory.