Distributional solutions of the stationary nonlinear Schrödinger equation: singularities, regularity and exponential decay

Abstract

We consider the nonlinear Schrödinger 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 $n\ge 3$ we use variational methods to prove that for all $p\in (\frac{n}{n-2},\frac{n}{n-2}+\eps)$ there exist distributional solutions with a point singularity at the origin provided $\eps>0$ is sufficiently small and $V,\Gamma$ are bounded on ${\mathbb{R}}^n\setminus B_1(0)$ and satisfy suitable Hölder-type conditions at the origin. In the case $n=1,2$ or $n\ge 3,1<p<\frac{n}{n-2}$, 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.