Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity

Abstract

We deal with a class on nonlinear Schrödinger equations (NLS) with potentials V(x) ∼ |x|-α, 0 < α < 2, and K(x) ∼ |x|-β, β> 0. Working in weighted Sobolev spaces, the existence of ground states uε belonging to W1,2(ℝ N) is proved under the assumption that σ < p