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The Lin-Ni's Problem for Mean Convex Domains

Memoirs of the American Mathematical Society

Format: Paperback / softback
Publisher: American Mathematical Society, Providence, United States
Published: 30th Jun 2012
Dimensions: w 178mm h 254mm d 13mm
Weight: 180g
ISBN-10: 0821869094
ISBN-13: 9780821869093
Barcode No: 9780821869093
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Synopsis
The authors 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, they show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\geq 7$. As a direct consequence, they 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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