Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials

Abstract

We study the existence of solutions of the following nonlinear Schrödinger equation−Δu+(V(x)−μ|x|2)u=f(x,u) for x∈RN∖{0}, where V:RN→R and f:RN×R→R are periodic in x∈RN. We assume that 0 does not lie in the spectrum of −Δ+V and μ<(N−2)24, N≥3. The superlinear and subcritical term f satisfies a weak monotonicity condition. For sufficiently small μ≥0 we find a ground state solution as a minimizer of the energy functional on a natural constraint. If μ<0 and 0 lies below the spectrum of −Δ+V, then ground state solutions do not exist.