Existence of standing waves for the nonlinear Schrödinger equation with double power nonlinearity and harmonic potential

Abstract

In this paper we prove the existence of standing waves for the nonlinear Schrödinger equation with double power nonlinearity and harmonic potential. The nonlinearity of our problem does not satisfy the global Ambrosetti-Rabinowitz condition. Therefore, in general, it seems difficult to obtain a boundedness of Palais-Smale sequence for the associated functional. We overcome this by the compactness argument.