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Let $\Omega\subset\BBR^N$ 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 Hardyexponents, $\gl_\gk$ the first eigenvalue of $\CL_\gk$ with correspondingpositive eigenfunction $\phi_\gk$. If $g$ is a continuous nondecreasingfunction satisfying$\int_1^\infty(g(s)+|g(-s)|)s^{-2\frac{2N-2+\ga_+}{2N-4+\ga_+}}ds<\infty$, thenfor any Radon measures $\gn\in \GTM_{\phi_\gk}(\Gw)$ and $\gm\in \GTM(\prt\Gw)$there exists a unique weak solution to problem $P_{\gn,\gm}$: $\CL_\gku+g(u)=\gn$ in $\Gw$, $u=\gm$ on $\prt\Gw$. If $g(r)=|r|^{q-1}u$ ($q>1$) weprove that, in the subcritical range of $q$, a necessary and sufficientcondition for solving $P_{0,\gm}$ with $\gm>0$ is that $\gm$ is absolutelycontinuous with respect to the capacity associated to the Besov space$B^{2-\frac{2+\ga_+}{2q'},q'}(\BBR^{N-1})$. We also characterize the boundaryremovable sets in terms of this capacity. In the subcritical range of $q$ weclassify the isolated singularities of positive solutions.

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