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 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.