Isolated boundary singularities of semilinear elliptic equations |
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Authors: | Marie-Fran?oise Bidaut-V��ron Augusto C Ponce Laurent V��ron |
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Institution: | 1. Laboratoire de Math??matiques et Physique Th??orique (UMR CNRS 6083), F??d??ration Denis Poisson, Universit?? Fran?ois Rabelais, 37200, Tours, France 2. Institut de Recherche en Math??matique et Physique, Universit?? Catholique de Louvain, Chemin du Cyclotron 2, 1348, Louvain-la-Neuve, Belgium
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Abstract: | Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 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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