Semilinear elliptic equations with Hardy potential and subcritical source term

TL;DR

This paper investigates positive solutions to semilinear elliptic equations with Hardy potential and subcritical source terms, establishing existence, nonexistence, and qualitative properties depending on parameters and boundary conditions.

Contribution

It introduces new results on existence and nonexistence of solutions for equations with Hardy potential, including a critical exponent and boundary trace analysis.

Findings

Existence of solutions for subcritical nonlinearities with small boundary measure.

Nonexistence of solutions for supercritical nonlinearities with isolated boundary singularities.

Extension of existence results to more general subcritical source functions.

Abstract

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ and $\delta(x)=\text{dist}\,(x,\partial \Omega)$. Assume $\mu>0$, $\nu$ is a nonnegative finite measure on $\partial \Omega$ and $g \in C(\Omega \times \mathbb{R}_+)$. We study positive solutions of $$ (P)\qquad -\Delta u - \frac{\mu}{\delta^2} u = g(x,u) \text{ in } \Omega, \qquad \text{tr}^*(u)=\nu. $$ Here $\text{tr}^*(u)$ denotes the normalized boundary trace of $u$ which was recently introduced by M. Marcus and P. T. Nguyen. We focus on the case $0<\mu < C_H(\Omega)$ (the Hardy constant for $\Omega$) and provide some qualitative properties of solutions of (P). When $g(x,u)=u^q$ with $q>1$, we prove that there is a critical value $q^*$ (depending only on $N$, $\mu$) for (P) in the sense that if $1<q<q^*$ then (P) admits a solution under a smallness assumption on $\nu$, but if $q \geq q^*$ this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where $g$ is subcritical. We also investigate the case where the $g$ is linear or sublinear and give some existence results for (P).