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

Shengbing Deng, Fethi Mahmoudi, Monica Musso

Research output: Contribution to journalArticle

Abstract

Let Ω be an open bounded domain in Rn with smooth boundary ∂Ω. We consider the equation ∆u + u n−k+2 n−k−2−ε = 0 in Ω, 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 |∇uε|2 * S n−k 2 n−k δK as ε → 0 where δK stands for the Dirac measure supported on K and Sn−k is an explicit positive constant. This result generalizes the one obtained in [17], where the case k = 1 is considered.

Original languageEnglish
JournalProceedings of the London Mathematical Society
DOIs
Publication statusAccepted/In press - 27 Jun 2018

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Minimal Submanifolds
Sectional Curvature
Weighted Average
Elliptic Problems
Submanifolds
Critical Exponents
Dirichlet Problem
Dirichlet Boundary Conditions
Paul Adrien Maurice Dirac
Bounded Domain
Closed
Generalise
Zero

Cite this

Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents. / Deng, Shengbing; Mahmoudi, Fethi; Musso, Monica.

In: Proceedings of the London Mathematical Society, 27.06.2018.

Research output: Contribution to journalArticle

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AB - Let Ω be an open bounded domain in Rn with smooth boundary ∂Ω. We consider the equation ∆u + u n−k+2 n−k−2−ε = 0 in Ω, 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 |∇uε|2 * S n−k 2 n−k δK as ε → 0 where δK stands for the Dirac measure supported on K and Sn−k is an explicit positive constant. This result generalizes the one obtained in [17], where the case k = 1 is considered.

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