Partial Differential Equations
Capacitary estimates of solutions of a class of nonlinear elliptic equations
Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 913-918.

Let Ω be a smooth bounded domain in N and K a compact subset of Ω. Assume that q⩾(N+1)/(N−1) and denote by UK the maximal solution of −Δu+uq=0 in Ω which vanishes on ΩK. We obtain sharp upper and lower estimates for UK in terms of the Bessel capacity C2/q,q and prove that UK is σ-moderate. In addition we relate the strong ‘blow-up’ points of UK on Ω to the ‘thick’ points of K in the fine topology associated with C2/q,q and characterize these points by a path integral condition on UK.

Soit Ω un domaine borné régulier de N and K un sous-ensemble compact de Ω. Supposons q⩾(N+1)/(N−1) et soit UK la solution maximale de ()-Δu+u q =0 dans Ω qui s'annulle sur ΩK. Nous obtenons des majorations et minorations précises de UK au moyen de la capacité de Bessel C2/q,q et montrons que UK est σ-modérée. En outre nous corrélons les points d'explosion forte de UK et les points épais de K pour la topologie fine associée à C2/q,q et caractérisons ces points par une condition d'intégrale de chemin portant sur UK.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00217-6
Marcus, Moshe 1; Véron, Laurent 2

1 Department of Mathematics, Israel Institute of Technology-Technion, 32000 Haifa, Israel
2 Département de mathématiques, Faculté des sciences et techniques, Université de Tours, 37200 Tours, France
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Marcus, Moshe; Véron, Laurent. Capacitary estimates of solutions of a class of nonlinear elliptic equations. Comptes Rendus. Mathématique, Volume 336 (2003) no. 11, pp. 913-918. doi : 10.1016/S1631-073X(03)00217-6. http://www.numdam.org/articles/10.1016/S1631-073X(03)00217-6/

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This research was supported by RTN contract No. HPRN-CT-2002-00274.