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arXiv · 0903.3301

Nonlinear Schrödinger equations with strongly singular potentials

Jacopo Bellazzini·Claudio Bonanno
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Abstract

In this paper we look for standing waves for nonlinear Schrödinger equations $$ i\frac{\partial ψ}{\partial t}+Δψ- g(|y|) ψ-W^{\prime}(| ψ|)\fracψ{| ψ|}=0 $$ 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.

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Jacopo Bellazzini, Claudio Bonanno. 2009-03-19. Nonlinear Schrödinger equations with strongly singular potentials. https://arxiv.org/abs/0903.3301

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