Measure boundary value problem for semilinear elliptic equations with critical Hardy potentials

Abstract

Let $Ω\subset\BBR^N$ be a bounded $C^2$ domain and $\CL_\gk=-\Gd-\frac{\gk}{d^2}$ the Hardy operator where $d=\dist (.,\prt\Gw)$ and $0<\gk\leq\frac{1}{4}$. Let $\ga_{\pm}=1\pm\sqrt{1-4\gk}$ be the two Hardy exponents, $\gl_\gk$ the first eigenvalue of $\CL_\gk$ with corresponding positive eigenfunction $ϕ_\gk$. If $g$ is a continuous nondecreasing function satisfying $\int_1^\infty(g(s)+|g(-s)|)s^{-2\frac{2N-2+\ga_+}{2N-4+\ga_+}}ds<\infty$, then for any Radon measures $\gn\in \GTM_{ϕ_\gk}(\Gw)$ and $\gm\in \GTM(\prt\Gw)$ there exists a unique weak solution to problem $P_{\gn,\gm}$: $\CL_\gk u+g(u)=\gn$ in $\Gw$, $u=\gm$ on $\prt\Gw$. If $g(r)=|r|^{q-1}u$ ($q>1$) we prove that, in the subcritical range of $q$, a necessary and sufficient condition for solving $P_{0,\gm}$ with $\gm>0$ is that $\gm$ is absolutely continuous with respect to the capacity associated to the Besov space $B^{2-\frac{2+\ga_+}{2q'},q'}(\BBR^{N-1})$. We also characterize the boundary removable sets in terms of this capacity. In the subcritical range of $q$ we classify the isolated singularities of positive solutions.