Capacitary estimates of solutions of a class of nonlinear elliptic equations

Marcus, Moshe; Véron, Laurent

Comptes Rendus. Mathématique, Tome 336 (2003) no. 11, pp. 913-918.

Soit $\Omega$ un domaine borné régulier de ${ℝ}^N$ and K un sous-ensemble compact de $\partial \Omega$. Supposons q⩾(N+1)/(N−1) et soit U_K la solution maximale de $\left(ℰ\right)-\Delta \mathrm{u}+{\mathrm{u}}^q=0$ dans $\Omega$ qui s'annulle sur $\partial \Omega ⧹\mathrm{K}$. Nous obtenons des majorations et minorations précises de U_K au moyen de la capacité de Bessel C_2/q,q′ et montrons que U_K est σ-modérée. En outre nous corrélons les points d'explosion forte de U_K et les points épais de K pour la topologie fine associée à C_2/q,q′ et caractérisons ces points par une condition d'intégrale de chemin portant sur U_K.

Let $\Omega$ be a smooth bounded domain in ${ℝ}^N$ and K a compact subset of $\partial \Omega$. Assume that q⩾(N+1)/(N−1) and denote by U_K the maximal solution of −Δu+u^q=0 in $\Omega$ which vanishes on $\partial \Omega ⧹\mathrm{K}$. 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 σ-moderate. In addition we relate the strong ‘blow-up’ points of U_K on $\partial \Omega$ to the ‘thick’ points of K in the fine topology associated with C_2/q,q′ and characterize these points by a path integral condition on U_K.

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