Nonlinear Schrodinger equations with strongly singular potentials

We look for standing waves for nonlinear Schrodinger equations i partial derivative psi/partial derivative t + Delta psi - g(vertical bar y vertical bar)psi - W'(vertical bar psi vertical bar) psi/vertical bar psi vertical bar = 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 a lack of compactness. As a specific case, we prove the existence of standing waves with non-vanishing angular momentum for the nonlinear hydrogen atom equation.