Standing waves for nonlinear Schrodinger equations with a general nonlinearity: One and two dimensional cases

Abstract

For N=1,2, we consider singularly perturbed elliptic equations epsilon(2) Delta u-V(x) u+f(u)=0, u(x)> 0 on R-N, lim(vertical bar x vertical bar ->infinity) u(x)=0. For small epsilon > 0, we show the existence of a localized bound state solution concentrating at an isolated component of positive local minimum of V under conditions on f we believe to be almost optimal; when N >= 3, it was shown in Byeon and Jeanjean (2007).