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

### Rainer Mandel

Karlsruhe Institute of Technology (KIT), Germany### Wolfgang Reichel

Karlsruhe Institute of Technology (KIT), Germany

## Abstract

We consider the nonlinear Schr\"{o}dinger equation

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 $n\geq 3,1<p<\frac{n}{n-2}$, however, we show that every distributional solution of the more general equation $-\Delta u + V(x) u = g(x,u)$ is a bounded strong solution if $V$ is bounded and $g$ satisfies certain growth conditions.

## Cite this article

Rainer Mandel, Wolfgang Reichel, Distributional Solutions of the Stationary Nonlinear Schrödinger Equation: Singularities, Regularity and Exponential Decay. Z. Anal. Anwend. 32 (2013), no. 1, pp. 55–82

DOI 10.4171/ZAA/1474