The Lin-Ni's Problem for Mean Convex Domains
Author | : Olivier Druet |
Publisher | : American Mathematical Soc. |
Total Pages | : 118 |
Release | : 2012 |
ISBN-10 | : 9780821869093 |
ISBN-13 | : 0821869094 |
Rating | : 4/5 (094 Downloads) |
Download or read book The Lin-Ni's Problem for Mean Convex Domains written by Olivier Druet and published by American Mathematical Soc.. This book was released on 2012 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.