Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Nonlinear Schrödinger equations with strongly singular potentials

Jacopo Bellazzinia1 and Claudio Bonannoa1

a1 Dipartimento di Matematica Applicata ‘U. Dini’, Università di Pisa, via Buonarroti 1/c, 56127 Pisa, Italy (j.bellazzini@ing.unipi.it; bonanno@mail.dm.unipi.it)

Abstract

We look for standing waves for nonlinear Schrödinger equations

$$\mathrm{i}\frac{\partial\psi}{\partial t}+\Delta\psi-g(|y|)\psi -W^{\prime}(|\psi|)\frac{\psi}{|\psi|}=0 $$

with cylindrically symmetric potentials g vanishing at infinity and non-increasing, and a C1 nonlinear term satisfying weak assumptions. In particular, we show the existence of standing waves with non-vanishing angular momentum with prescribed L2 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.

(Received September 07 2009)

(Accepted November 09 2009)