The Lin-Ni’s problem for mean convex domains

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...

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Bibliographic Details
Main Authors: Druet, Olivier, Robert, Frédéric, Wei, Juncheng
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2011
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Blowing up (Algebraic geometry)
Convex domains
Differential equations, Elliptic
Neumann problem
Neumann elliptic problem
critical exponent
blow-up
ISBN:9780821869093, 0821869094
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Summary: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.
Bibliography:"July 2012, volume 218, number 1027 (end of volume)"--T.p
Includes bibliography (p. 103-105)
ISBN:9780821869093
0821869094
ISSN:0065-9266
1947-6221
DOI:10.1090/S0065-9266-2011-00646-5