The Lin-Ni's problem for mean convex domains

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We prove some refined asymptotic estimates for postive blowing up solutions to Δu+εu=n(n-2)uⁿ⁺²/ⁿ⁻² on Ω, ∂νu=0 on ∂Ω; Ω being a smooth bounded domain of, n≥ 3. In particular, we show that concentration can occur only on boundary points with nonpositive mean curvature when n=3 or n≥ 7. As a direct consequence, we 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.