JournalsjemsVol. 6 , No. 4DOI 10.4171/jems/18

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

Abstract

Let \Gw\Gw be a bounded domain of class C2C^2 in RN\mathbb R^N and let KK be a compact subset of \prt\Gw\prt\Gw. Assume that q(N+1)/(N1)q\geq (N+1)/(N-1) and denote by UKU_K the maximal solution of \Gdu+uq=0-\Gd u+u^q=0 in \Gw\Gw which vanishes on \prt\GwK\prt\Gw\setminus K. We obtain sharp upper and lower estimates for UKU_K in terms of the Bessel capacity C2/q,qC_{2/q,q'} and prove that UKU_K is \gs\gs-moderate. In addition we describe the precise asymptotic behavior of UKU_K at points \gsK\gs\in K, which depends on the 'density' of KK at \gs\gs, measured in terms of the capacity C2/q,qC_{2/q,q'}.