Semilinear elliptic Schr\"odinger equations involving singular potentials and source terms

TL;DR

This paper investigates semilinear elliptic Schrödinger equations with singular potentials and measure data, providing comprehensive conditions for the existence of solutions based on advanced kernel estimates and capacity theories.

Contribution

It introduces new existence criteria for solutions to equations with inverse-square potentials and measure data, using refined potential theory techniques.

Findings

Established necessary and sufficient conditions for solutions.

Developed detailed estimates of Green and Martin kernels.

Analyzed the impact of singular potentials and measure data on solvability.

Abstract

Let $\Omega \subset \mathbb{R}^N$ ($N>2$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in $\Omega \setminus \Sigma$, where $d_\Sigma(x) = \mathrm{dist}(x,\Sigma)$ and $\mu$ is a parameter. We study the boundary value problem (P) $-L_\mu u = g(u) + \tau$ in $\Omega \setminus \Sigma$ with condition $u=\nu$ on $\partial \Omega \cup \Sigma$, where $g: \mathbb{R} \to \mathbb{R}$ is a nondecreasing, continuous function and $\tau$ and $\nu$ are positive measures. The interplay between the inverse-square potential $d_\Sigma^{-2}$, the nature of the source term $g(u)$ and the measure data $\tau,\nu$ yields substantial difficulties in the research of the problem. We perform a deep analysis based on delicate estimate on the Green kernel and Martin kernel and fine topologies induced by appropriate capacities to establish various necessary and sufficient conditions for the existence of a solution in different cases.