Semilinear elliptic equations with Hardy potential and gradient nonlinearity

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) be a $C^2$ bounded domain and $\delta$ be the distance to $\mathrm{\partial }\mathrm{\Omega }$. We study positive solutions of equation (E) $-L_{\mu }u+g(\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })=0$ in $\mathrm{\Omega }$ 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 $\mathrm{\partial }\mathrm{\Omega }$. In the supercritical case, i.e. $q_{\mu }\le q<2$, we demonstrate a removability result in terms of Bessel capacities.