Isolated boundary singularities of semilinear elliptic equations

Bidaut-Véron, Marie-Françoise; Ponce, Augusto; Véron, Laurent
January 2011
Calculus of Variations & Partial Differential Equations;Jan2011, Vol. 40 Issue 1/2, p183
Academic Journal
Given a smooth domain $${\Omega\subset\mathbb{R}^N}$$ such that $${0 \in \partial\Omega}$$ and given a nonnegative smooth function ζ on ∂Ω, we study the behavior near 0 of positive solutions of −Δ u = u in Ω such that u = ζ on ∂Ω\{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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