Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Abstract

Let $\Omega$ be a bounded domain of class $C^2$ in ${\mathbb{R}}^N$ and let $K$ be a compact subset of $\partial \Omega$. Assume that $q\ge (N+1)/(N-1)$ and denote by $U_K$ the maximal solution of $-\Delta u+u^q=0$ in $\Omega$ which vanishes on $\partial \Omega \setminus K$. 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 $\sigma$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\sigma \in K$, which depends on the “density” of $K$ at $\sigma$, measured in terms of the capacity $C_{2/q,q^{\prime }}$.