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arXiv · 1103.3811

The Lin-Ni's problem for mean convex domains

Olivier Druet·Frédéric Robert·Juncheng Wei
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Abstract

We prove some refined asymptotic estimates for postive blowing up solutions to $Δu+εu=n(n-2)u^{\frac{n+2}{n-2}}$ on $Ω$, $\partial_νu=0$ on $\partialΩ$; $Ω$ being a smooth bounded domain of $\rn$, $n\geq 3$. In particular, we show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\geq 7$. As a direct consequence, we prove the validity of the Lin-Ni's conjecture in dimension $n=3$ 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.

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Olivier Druet, Frédéric Robert, Juncheng Wei. 2011-03-19. The Lin-Ni's problem for mean convex domains. https://arxiv.org/abs/1103.3811

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