Semilinear elliptic equations with Hardy potential and gradient nonlinearity

Abstract

Let $\Omega \subset {\mathbb{R}}^N$ ($N\ge 3$) be a $C^2$ bounded domain, and let $\delta$ be the distance to $\partial \Omega$. In this paper, we study positive solutions of the equation $(\star )-L_{\mu }u+g(|\nabla u|)=0$ in $\Omega$, where $L_{\mu }=\Delta +\mu /{\delta }^2$ , $\mu \in (0,1/4]$ and $g$ is a continuous, nondecreasing function on ${\mathbb{R}}_{+}$. We prove that if $g$ satisfies a singular integral condition, then there exists a unique solution of $(\star )$ with a prescribed boundary datum $\nu$. When $g(t)=t^q$ with $q\in (1,2)$, we show that equation $(\star )$ admits a critical exponent $q_{\mu }$ (depending only on $N$ and $\mu$). In the subcritical case, namely $1<q<q_{\mu }$, we establish some a priori estimates and provide a description of solutions with an isolated singularity on $\partial \Omega$. In the supercritical case, i.e., $q_{\mu }\le q<2$, we demonstrate a removability result in terms of Bessel capacities.