Semilinear elliptic equations with Hardy potential and subcritical source term

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 $$(P)-\Delta u-\frac{\mu }{{\delta }^2}u=g(x,u)\text{ in }\Omega ,{\text{tr}}^{\ast }(u)=\nu .$$ Here ${\text{tr}}^{\ast }(u)$ 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 $q^{\ast }$ (depending only on $N$, $\mu$) for (P) in the sense that if $1<q<q^{\ast }$ then (P) admits a solution under a smallness assumption on $\nu$, but if $q\ge q^{\ast }$ 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).

arXiv comment27 pages