Semilinear elliptic Schr\"odinger equations with singular potentials and absorption terms

TL;DR

This paper studies semilinear elliptic Schrödinger equations with singular potentials and absorption terms, establishing sharp conditions for the existence of solutions, critical exponents, and capacity-based solvability criteria.

Contribution

It introduces new sharp growth conditions for the absorption term and identifies critical exponents for solution existence in singular potential problems.

Findings

Existence of solutions depends on growth conditions of g.

Identification of critical exponents for power-type g.

Capacity conditions characterize solvability.

Abstract

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) 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 investigate 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 complex interplay between the competing effects of the inverse-square potential $d_\Sigma^{-2}$, the absorption term $g(u)$ and the measure data $\tau,\nu$ discloses different scenarios in which problem (P) is solvable. We provide sharp conditions on the growth of $g$ for the existence of solutions. When $g$ is a power function, namely $g(u)=|u|^{p-1}u$ with $p>1$, we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e. $p$ is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e. $p$ is greater than a critical exponent). Finally, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities for the solvability of (P).