Semilinear Schr\"odinger equations with Hardy potentials involving the distance to a boundary submanifold and gradient source nonlinearities

TL;DR

This paper investigates positive solutions to a class of semilinear Schrödinger equations with Hardy potentials and gradient nonlinearities in bounded domains, establishing existence results under subcritical and certain supercritical conditions with boundary measure data.

Contribution

It introduces new existence criteria for solutions involving Hardy potentials and gradient nonlinearities, especially for supercritical exponents, using capacity-based conditions.

Findings

Existence of solutions under subcritical conditions.

Criteria for solutions with power-type nonlinearities.

Solutions exist when boundary measure data is small and satisfies capacity conditions.

Abstract

Let $\Omega\subset\mathbb{R}^N$ ($N\geq 3$) be a bounded $C^2$ domain and $\Sigma\subset\partial\Omega$ be a compact $C^2$ submanifold of dimension $k$. Denote the distance from $\Sigma$ by $d_\Sigma$. In this paper, we study positive solutions of the equation $(*)\, -\Delta u -\mu u/d_\Sigma^2 = g(u,|\nabla u|)$ in $\Omega$, where $\mu\leq \big( \frac{N-k}{2} \big)^2$ and the source term $g:\mathbb{R}\times\mathbb{R}_+ \rightarrow \mathbb{R}_+$ is continuous and non-decreasing in its arguments with $g(0,0)=0$. In particular, we prove the existence of solutions of $(*)$ with boundary measure data $u=\nu$ in two main cases, provided that the total mass of $\nu$ is small. In the first case $g$ satisfies some subcriticality conditions that always ensure the existence of solutions. In the second case we examine power type nonlinearity $g(u,|\nabla u|) = |u|^p|\nabla u|^q$, where the problem may not possess a solution for exponents in the supercritical range. Nevertheless we obtain criteria for existence under the assumption that $\nu$ is absolutely continuous with respect to some appropriate capacity or the Bessel capacity of $\Sigma$, or under other equivalent conditions.