Olivier Druet | Akateeminen Kirjakauppa

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 $partialOmega$, $Omega$ being a smooth bounded domain of $mathbb{R}^n$, $ngeq 3$. In particular, they show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $ngeq 7$. As a direct consequence, they prove the validity of the Lin-Ni's conjecture in dimension $n=3$ and $ngeq 7$ 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.