Weak solutions of semilinear elliptic equations with Leray-Hardy potential and measure data

We study existence and stability of solutions of (E 1) –$\mathrm{\Delta }$u + $\mu$ |x| 2 u + g(u) = $\nu$ in $\mathrm{\Omega }$, u = 0 on $\mathrm{\partial }$$\mathrm{\Omega }$, where $\mathrm{\Omega }$ is a bounded, smooth domain of R N , N $\ge$ 2, containing the origin, $\mu$ $\ge$ – (N --2) 2 4 is a constant, g is a nondecreasing function satisfying some integral growth assumption and $\nu$ is a Radon measure on $\mathrm{\Omega }$. We show that the situation differs according $\nu$ is diffuse or concentrated at the origin. When g is a power we introduce a capacity framework to find necessary and sufficient condition for solvability.