The Lin-Ni’s problem for mean convex domains

Olivier Druet (1) , Juncheng Wei (2) , Frédéric Robert (3)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 UBC - University of British Columbia
3 IECN - Institut Élie Cartan de Nancy

We prove some refined asymptotic estimates for postive blowing up solutions to ∆u+u = n(n−2)u n+2 n−2 on Ω, ∂ν u = 0 on ∂Ω; Ω being a smooth bounded domain of R n , 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 [32] show that the bound on the energy is a necessary condition. Frédéric Robert dedicates this work to Clémence Climaque

Domaines: Equations aux dérivées partielles [math.AP]

Olivier Druet, Juncheng Wei, Frédéric Robert. The Lin-Ni’s problem for mean convex domains. Memoirs of the American Mathematical Society, 2012, 218 (1027), ⟨10.1090/S0065-9266-2011-00646-5⟩. ⟨hal-01279343⟩