Maximal solutions of the equation $ \mathrm{\Delta }u={u}^{q}$ in arbitrary domains

Moshe Marcus; Laurent Véron

doi:10.1016/j.crma.2007.01.002

Partial Differential Equations

Maximal solutions of the equation

Δ u = u^{q}

in arbitrary domains

Presented by: Haïm Brezis

Moshe Marcus ¹; Laurent Véron 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

Comptes Rendus. Mathématique, Volume 344 (2007) no. 5, pp. 299-304.

Abstract
Résumé

We prove bilateral capacitary estimates for the maximal solution $U_{F}$ of $- Δ u + u^{q} = 0$ in the complement of an arbitrary closed set $F \subset R^{N}$ , involving the Bessel capacity $C_{2, q^{'}}$ , for q in the supercritical range $q ⩾ q_{c} : = N / (N - 2)$ . We derive a pointwise necessary and sufficient condition, via a Wiener type criterion, in order that $U_{F} (x) \to \infty$ as $x \to y$ for given $y \in \partial F$ . Finally we prove a general uniqueness result for large solutions.

Nous démontrons une estimation capacitaire bilatérale de la solution maximale $U_{F}$ de $- Δ u + u^{q} = 0$ dans un domaine quelconque de $R^{N}$ impliquant la capacité de Bessel $C_{2, q^{'}}$ dans le cas sur-critique $q ⩾ q_{c} : = N / (N - 2)$ . 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: 2007-01-01
Published online: 2007-02-09

DOI: 10.1016/j.crma.2007.01.002

Author's affiliations:

Moshe Marcus ¹; Laurent Véron 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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     author = {Moshe Marcus and Laurent V\'eron},
     title = {Maximal solutions of the equation $ \mathrm{\Delta }u={u}^{q}$ in arbitrary domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {299--304},
     publisher = {Elsevier},
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     number = {5},
     year = {2007},
     doi = {10.1016/j.crma.2007.01.002},
     language = {en},
}

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Moshe Marcus; Laurent Véron. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.01.002/

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