The Lin-Ni's problem for mean convex domains

We prove some refined asymptotic estimates for postive blowing up solutions to $\mathrm{\Delta }u+ϵu=n(n-2)u^{\frac{n+2}{n-2}}$ on $\mathrm{\Omega }$, $\partial_\nu u=0$on$\partial\Omega$;$\Omega$beingasmoothboundeddomainof$\rn$,$n\geq 3$. In particular, we show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\ge 7$. As a direct consequence, we prove the validity of the Lin-Ni’s conjecture in dimension $n=3$ and $n\ge 7$ 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.