Nonlinear Schrödinger equations with strongly singular potentials

Abstract

In this paper we look for standing waves for nonlinear Schrödinger equations $$i\frac{\partial \psi }{\partial t}+\Delta \psi -g(|y|)\psi -W^{\prime }(|\psi |)\frac{\psi }{|\psi |}=0$$ with cylindrically symmetric potentials $g$ vanishing at infinity and non-increasing, and a $C^1$ nonlinear term satisfying weak assumptions. In particular we show the existence of standing waves with non-vanishing angular momentum with prescribed $L^2$ norm. The solutions are obtained via a minimization argument, and the proof is given for an abstract functional which presents lack of compactness. As a particular case we prove the existence of standing waves with non-vanishing angular momentum for the nonlinear hydrogen atom equation.