Semilinear elliptic equations with Hardy potential and subcritical source term

Abstract

Let $Ω$ be a smooth bounded domain in $\mathbb{R}^N$ and $δ(x)=\text{dist}\,(x,\partial Ω)$. Assume $μ>0$, $ν$ is a nonnegative finite measure on $\partial Ω$ and $g \in C(Ω\times \mathbb{R}_+)$. We study positive solutions of $$ (P)\qquad -Δu - \fracμ{δ^2} u = g(x,u) \text{ in } Ω, \qquad \text{tr}^*(u)=ν. $$ 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 $0<μ< C_H(Ω)$ (the Hardy constant for $Ω$) 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 $N$, $μ$) for (P) in the sense that if $1<q<q^*$ then (P) admits a solution under a smallness assumption on $ν$, 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).