The Lin-Ni's problem for mean convex domains

TL;DR

This paper refines asymptotic estimates for positive solutions to a nonlinear PDE on smooth bounded domains, establishing boundary concentration conditions and confirming Lin-Ni's conjecture in specific dimensions.

Contribution

It provides new asymptotic estimates and proves Lin-Ni's conjecture for mean convex domains in dimensions 3 and ≥7, clarifying boundary concentration behavior.

Findings

Concentration occurs only at boundary points with nonpositive mean curvature in specified dimensions.

Lin-Ni's conjecture holds for mean convex domains in dimensions 3 and ≥7.

Bounded energy is necessary for the conjecture's validity.

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 $\rn$, $n\geq 3$. In particular, we show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\geq 7$. As a direct consequence, we prove the validity of the Lin-Ni's conjecture in dimension $n=3$ and $n\geq 7$ 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.