The Lin-Ni’s problem for mean convex domains - Druet, Olivier; Robert, Frédéric; Wei, Juncheng (Memoirs of the American Mathematical Society, 2011)
The authors prove some refined asymptotic estimates for positive blow-up solutions to $Δ u+ε u=n(n-2)u^{\frac{n+2}{n-2}}$ on $Ω$, $\partial_ν u=0$ on $\partialΩ$, $Ω$ being a smooth bounded domain of $\mathbb{R}^n$, $n≥ 3$. In particular, they show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n≥ 7$. As a direct consequence, they prove the validity of the Lin-Ni's conjecture in dimension $n=3$ and $n≥ 7$ 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.