Nonlinear Schrödinger equations with strongly singular potentials

2009/03/19 by Jacopo Bellazzini, Claudio Bonanno, Bellazzini, Jacopo +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.0903.3301

arxiv created 2009/03/19 · arxiv updated 2009/12/01

Abstract

In this paper we look for standing waves for nonlinear Schrödinger equations i(∂ ψ)/(∂ t)+Δψ- g(|y|) ψ-W^′(| ψ|)\fracψ| ψ|=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 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.

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