Capacitary estimates of positive solutions of semilinear elliptic equations with absorption
Summary: 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 \(q\geq (N+1)/(N-1)\) 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 \(C_{2/q,q'}\) 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 \(C_{2/q,q'}\).
Bessel capacities, maximal solutions, rate of blow-up
scientific article; zbMATH DE number 2139681