The Lin-Ni’s problem for mean convex domains

Druet, Olivier; Robert, Frédéric; Wei, Juncheng

doi:10.1090/S0065-9266-2011-00646-5

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The Lin-Ni's problem for mean convex domains

Authors: Olivier Druet, Frédéric Robert and Juncheng Wei
Journal: Memoirs of the AMS 218 (2012), no. 1027
MSC (2010): Primary 35J20, 35J60
Posted: November 30, 2011
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Abstract: We prove some refined asymptotic estimates for positive blow-up solutions to $\Delta u+\epsilon u=n(n-2)u^{\frac {n+2}{n-2}}$ on $\Omega$ , $\partial _\nu u=0$ on $\partial \Omega$ , $\Omega$ being a smooth bounded domain of $\mathbb{R}^n$ , $n\geq 3$ . In particular, we show that concentration can occur only on boundary points with nonpositive mean curvature when or $n\geq 7$ . As a direct consequence, we prove the validity of the Lin-Ni's conjecture in dimension and $n\geq 7$ for mean convex domains and with bounded energy. Recent examples by Wang-Wei-Yan show that the bound on the energy is a necessary condition.

Frédéric Robert dedicates this work to Clémence Climaque

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