Partial Differential Equations
Maximal solutions of the equation Δu=uq in arbitrary domains
Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 299-304.

We prove bilateral capacitary estimates for the maximal solution UF of Δu+uq=0 in the complement of an arbitrary closed set FRN, involving the Bessel capacity C2,q, for q in the supercritical range qqc:=N/(N2). We derive a pointwise necessary and sufficient condition, via a Wiener type criterion, in order that UF(x) as xy for given yF. Finally we prove a general uniqueness result for large solutions.

Nous démontrons une estimation capacitaire bilatérale de la solution maximale UF de Δu+uq=0 dans un domaine quelconque de RN impliquant la capacité de Bessel C2,q dans le cas sur-critique qqc:=N/(N2). Grâce à un critère de type Wiener, nous en déduisons une condition nécessaire et suffisante pour que cette solution maximale tende vers l'infini en un point du bord du domaine. Finalement nous prouvons un résultat général d'unicité des grandes solutions.

Accepted:
Published online:
DOI: 10.1016/j.crma.2007.01.002
Marcus, Moshe 1; Véron, Laurent 2

1 Department of Mathematics, Technion, Haifa 32000, Israel
2 Laboratoire de mathématiques et physique théorique, faculté des sciences, parc de Grandmont, 37200 Tours, France
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Marcus, Moshe; Véron, Laurent. Maximal solutions of the equation $ \mathrm{\Delta }u={u}^{q}$ in arbitrary domains. Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 299-304. doi : 10.1016/j.crma.2007.01.002. http://www.numdam.org/articles/10.1016/j.crma.2007.01.002/

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