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 $\delta$ be the distance to $\partial \Omega$. We study positive solutions of equation (E) $-L_{\mu }u+g(|\nabla u|)=0$ in $\Omega$ where $L_{\mu }=\Delta +\frac{\mu }{{\delta }^2}$, $\mu \in (0,\frac{1}{4}]$ 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 (E) with a prescribed boundary datum $\nu$. When $g(t)=t^q$ with $q\in (1,2)$, we show that equation (E) 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.

43 pages