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We study the boundary behaviour of the of (E) $-\Gd u-\myfrac{\xk}{d^2(x)}u+g(u)=0$, where $0<\xk <\frac{1}{4}$ and $g$ is a continuousnonndecreasing function in a bounded convex domain of $\BBR^N$. We firstconstruct the Martin kernel associated to the the linear operator $\CL_{\xk}=-\Gd-\frac{\xk }{d^2(x)}$ and give a general condition for solving equation(E) with any Radon measure $\gm$ for boundary data. When $g(u)=|u|^{q-1}u$ weshow the existence of a critical exponent $q_c=q_c(N,\xk )>1$: when $0<q<q_c$any measure is eligible for solving (E) with $\gm$ for boundary data; if $q\geqq_c$, a necessary and sufficient condition is expressed in terms of theabsolute continuity of $\gm$with respect to some Besov capacity. The samecapacity characterizes the removable compact boundary sets. At end any positivesolution (F) $-\Gd u-\frac{\xk }{d^2(x)}u+|u|^{q-1}u=0$ with $q>1$ admits aboundary trace which is a positive outer regular Borel measure. When $1<q<q_c$we prove that to any positive outer regular Borel measure we can associate apositive solutions of ($F$) with this boundary trace.

Boundary singularities of solutions of semilinear elliptic equations with critical Hardy potentials