A positive solution of a nonlinear Schrödinger equation with GG-symmetry

In this paper we consider the following nonlinear Schrödinger equations: {−Δu+V(x)u=f(u)in RN,u∈H1(RN). Here N≥2N≥2 and V(x)∈C(RN,R) is invariant under a group action G⊂O(N)G⊂O(N). We deal with both finite group actions and infinite group actions. The existence of a positive solution will be shown for a wide class of nonlinearities f(u)f(u).