Nonlinear Schr\"odinger equations with strongly singular potentials
Bellazzini, Jacopo ; Bonanno, Claudio
arXiv, 0903.3301 / Harvested from arXiv
Text on arXiv
In this paper we look for standing waves for nonlinear Schr\"odinger equations $$ i\frac{\partial \psi}{\partial t}+\Delta \psi - g(|y|) \psi -W^{\prime}(| \psi |)\frac{\psi}{| \psi |}=0 $$ 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.
Publié le : 2009-03-19
Classification:  Mathematical Physics 
@article{0903.3301,
     author = {Bellazzini, Jacopo and Bonanno, Claudio},
     title = {Nonlinear Schr\"odinger equations with strongly singular potentials},
     journal = {arXiv},
     volume = {2009},
     number = {0},
     year = {2009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0903.3301}
}

Bellazzini, Jacopo; Bonanno, Claudio. Nonlinear Schr\"odinger equations with strongly singular potentials. arXiv, Tome 2009 (2009) no. 0, . http://gdmltest.u-ga.fr/item/0903.3301/