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Pré-Publication, Document De Travail Année : 2009

ISOLATED BOUNDARY SINGULARITIES OF SEMILINEAR ELLIPTIC EQUATIONS

Résumé

Given a smooth domain $\Omega\subset\RR^N$ such that $0 \in \partial\Omega$ and given a nonnegative smooth function $\zeta$ on $\partial\Omega$, we study the behavior near $0$ of positive solutions of $-\Delta u=u^q$ in $\Omega$ such that $u = \zeta$ on $\partial\Omega\setminus\{0\}$. We prove that if $\frac{N+1}{N-1} < q < \frac{N+2}{N-2}$, then $u(x)\leq C \abs{x}^{-\frac{2}{q-1}}$ and we compute the limit of $\abs{x}^{\frac{2}{q-1}} u(x)$ as $x \to 0$. We also investigate the case $q= \frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.
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Dates et versions

hal-00358178 , version 1 (03-02-2009)
hal-00358178 , version 2 (09-07-2009)
hal-00358178 , version 3 (14-07-2009)

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Citer

Marie-Françoise Bidaut-Veron, Augusto C. Ponce, Laurent Veron. ISOLATED BOUNDARY SINGULARITIES OF SEMILINEAR ELLIPTIC EQUATIONS. 2009. ⟨hal-00358178v1⟩
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