G-invariant positive solutions for a class of locally superlinear Schrödinger equations

Abstract

In this paper, we are concerned with a nonlinear Schrödinger equation whose nonlinear term contains a positive parameter λ and has a superlinear growth only locally. Moreover, we are interested in the situation where the potential V(x) is invariant under a finite group action G, which enables us to treat the case that V(x) is of non-trapping type. By considering several auxiliary problems, we are able to obtain the existence of a G-invariant positive solution when the parameter λ is sufficiently large.