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Zeitschrift für Analysis und ihre Anwendungen


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Volume 32, Issue 1, 2013, pp. 55–82
DOI: 10.4171/ZAA/1474

Published online: 2013-01-27

Distributional Solutions of the Stationary Nonlinear Schrödinger Equation: Singularities, Regularity and Exponential Decay

Rainer Mandel[1] and Wolfgang Reichel[2]

(1) Karlsruhe Institute of Technology (KIT), Germany
(2) Karlsruhe Institute of Technology (KIT), Germany

We consider the nonlinear Schr\"{o}dinger equation $$-\Delta u + V(x) u = \Gamma(x) |u|^{p-1}u$$ in $\mathbb R^n$ where the spectrum of $-\Delta+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in \big(\frac{n}{n-2},\frac{n}{n-2}+\epsilon \big)$ there exist distributional solutions with a point singularity at the origin provided $\epsilon >0$ is sufficiently small and $V,\Gamma$ are bounded on $\mathbb R^n\setminus B_1(0)$ and satisfy suitable H\"{o}lder-type conditions at the origin. In the case $n=1,2$ or $n\geq 3,1

Keywords: Nonlinear Schrödinger equation, singular solutions, variational methods, distributional solutions

Mandel Rainer, Reichel Wolfgang: Distributional Solutions of the Stationary Nonlinear Schrödinger Equation: Singularities, Regularity and Exponential Decay. Z. Anal. Anwend. 32 (2013), 55-82. doi: 10.4171/ZAA/1474