# 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 $R_{n}$ where the spectrum of $−Δ+V(x)$ is positive. In the case $n≥3$ we use variational methods to prove that for all $p∈(n−2n ,n−2n +ϵ)$ there exist distributional solutions with a point singularity at the origin provided $ϵ>0$ is sufficiently small and $V,Γ$ are bounded on $R_{n}∖B_{1}(0)$ and satisfy suitable H\"{o}lder-type conditions at the origin. In the case $n=1,2$ or $n≥3,1<p<n−2n $, however, we show that every distributional solution of the more general equation $−Δu+V(x)u=g(x,u)$ 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