Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents

Abstract

Let normalΩ be an open bounded domain in Rn with smooth boundary ∂Ω. We consider the equation Δu+un−k+2n−k−2−ε=0inΩ, under zero Dirichlet boundary condition, where ε is a small positive parameter. We assume that there is a k‐dimensional closed, embedded minimal sub‐manifold K of ∂Ω, which is non‐degenerate, and along which a certain weighted average of sectional curvatures of ∂Ω is negative. Under these assumptions, we prove existence of a sequence ε=εj and a solution uε which concentrate along K, as ε→0+, in the sense that false|∇uε|2⇀Sn−kn−k2δKasε→0,where δK stands for the Dirac measure supported on K and Sn−k is an explicit positive constant.