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

TL;DR

This paper investigates the boundary value problem for semilinear elliptic equations with critical Hardy potentials, establishing existence, uniqueness, and boundary behavior of solutions under various conditions.

Contribution

It introduces new existence and uniqueness results for boundary value problems involving Hardy potentials and nonlinearities, with a detailed capacity-based characterization of solutions.

Findings

Existence and uniqueness of solutions for measure data with Hardy potentials.

Capacity conditions for solvability in the subcritical nonlinear range.

Characterization of removable boundary sets and singularities.

Abstract

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 Hardy exponents, $\gl_\gk$ the first eigenvalue of $\CL_\gk$ with corresponding positive eigenfunction $\phi_\gk$. 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 $\gn\in \GTM_{\phi_\gk}(\Gw)$ 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.