Standing Waves for Nonlinear Schrodinger Equations with a General Nonlinearity: One and Two Dimensional Cases.

For N = 1,2, we consider singularly perturbed elliptic equations ε2Δ u - V(x) u + f(u)= 0, u(x)> 0 on RN, lim|x|→∞u(x)= 0. For small ε > 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). [ABSTRACT FROM AUTHOR]