Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Abstract

Let \( \Gw \) be a bounded domain of class $C^2$ in ${\mathbb{R}}^N$ and let $K$ be a compact subset of \( \prt\Gw \). Assume that $q\ge (N+1)/(N-1)$ and denote by $U_K$ the maximal solution of \( -\Gd u+u^q=0 \) in \( \Gw \) which vanishes on \( \prt\Gw\setminus K \). We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q^{\prime }}$ and prove that $U_K$ is \( \gs \)-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points \( \gs\in K \), which depends on the 'density' of $K$ at \( \gs \), measured in terms of the capacity $C_{2/q,q^{\prime }}$.