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

TL;DR

This paper investigates the boundary behavior of solutions to a class of semilinear elliptic equations with critical Hardy potentials, establishing conditions for existence, uniqueness, and boundary trace characterization based on capacity theory.

Contribution

It constructs the Martin kernel for the linear operator with Hardy potential and characterizes boundary measures for solutions, introducing a critical exponent and capacity conditions.

Findings

Existence of a critical exponent q_c depending on N and k.

Boundary measure solvability depends on absolute continuity with respect to a Besov capacity.

Positive solutions have well-defined boundary traces as measures.

Abstract

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 continuous nonndecreasing function in a bounded convex domain of $\BBR^N$. We first construct 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 $g(u)=|u|^{q-1}u$ we show 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 $q\geq q_c$, a necessary and sufficient condition is expressed in terms of the absolute continuity of $\gm$with respect to some Besov capacity. The same capacity characterizes the removable compact boundary sets. At end any positive solution (F) $-\Gd u-\frac{\xk }{d^2(x)}u+|u|^{q-1}u=0$ with $q>1$ admits a boundary 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 a positive solutions of ($F$) with this boundary trace.