In this paper we look for standing waves for nonlinear Schr\"odinger equations $$ i\frac{\partial \psi }{\partial t}+\Delta \psi - g(|y|) \psi -W^{\prime }(\left| \psi \right| )\frac{\psi }{\left| \psi \right| }=0 $$ 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.
Nonlinear Schroedinger equations with strongly singular potentials
BELLAZZINI J;BONANNO, CLAUDIO
2010-01-01
Abstract
In this paper we look for standing waves for nonlinear Schr\"odinger equations $$ i\frac{\partial \psi }{\partial t}+\Delta \psi - g(|y|) \psi -W^{\prime }(\left| \psi \right| )\frac{\psi }{\left| \psi \right| }=0 $$ 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.