The Lin-Ni's problem for mean convex domains

Abstract

We prove some refined asymptotic estimates for postive blowing 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 $\rn$, $n\ge 3$. 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.

To appear in "Memoirs of the AMS"

References in corpus (1)

• Asymptotic analysis for fourth order Paneitz equations with critical growth

Cited by in corpus (6)

• Non-existence of extremals for the Adimurthi-Druet inequality
• Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method
• On Lin-Ni's conjecture in dimensions four and six
• Blow-up analysis involving isothermal coordinates on the boundary of compact Riemann surface
• Hardy-Sobolev equations with asymptotically vanishing singularity: Blow-up analysis for the minimal energy
• Sur les estimations du type sup+inf et sup*inf