Distributional solutions of the stationary nonlinear Schr\"odinger equation: singularities, regularity and exponential decay

TL;DR

This paper investigates distributional solutions to the stationary nonlinear Schrödinger equation, establishing existence of solutions with singularities in higher dimensions and regularity results in lower dimensions under various conditions.

Contribution

It introduces new existence results for singular solutions in higher dimensions and proves regularity of solutions in lower dimensions for a broad class of nonlinearities.

Findings

Existence of distributional solutions with point singularities in dimensions n≥3 for certain p-values.

All distributional solutions are bounded and regular under specified growth conditions in dimensions 1, 2, and for certain p in higher dimensions.

Solutions exhibit exponential decay and regularity depending on the dimension and nonlinear growth conditions.

Abstract

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