# Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

### Moshe Marcus

Technion - Israel Institute of Technology, Haifa, Israel### Laurent Véron

Université François Rabelais, Tours, France

## 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 $q\geq (N+1)/(N-1)$ 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 $C_{2/q,q'}$ 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 $C_{2/q,q'}$.