Maximal solutions of equation u = uq in arbitrary domains

Details Maximal solutions of equation u = uq in arbitrary domains Maximal solutions of equation u = uq in arbitrary domains

2008-05-24

arxiv.org/abs/0805.3787v1

Comptes Rendus de l Acad\'emie des Sciences - Series I - Mathematics 344 (2007) 299-304

Mathematics

Analysis of PDEs

Scientific paper

We prove bilateral capacitary estimates for the maximal solution $U_F$ of $-\Delta u+u^q=0$ in the complement of an arbitrary closed set $F\subset\mathbb R^N$, involving the Bessel capacity $C_{2,q'}$, for $q$ in the supercritical range $q\geq q_{c}:=N/(N-2)$. 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 $y\in\prt F$. Finally we prove a general uniqueness result for large solutions.

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