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Abstract
Let Ω be an open bounded domain in R^{n} 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 kdimensional closed, embedded minimal submanifold K of ∂Ω, which is nondegenerate, 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 u_{ε} which concentrate along K, as ε → 0+, in the sense that ∇u_{ε}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 language  English 

Pages (fromto)  379415 
Number of pages  37 
Journal  Proceedings of the London Mathematical Society 
Volume  118 
Issue number  2 
Early online date  2 Aug 2018 
DOIs  
Publication status  Published  1 Feb 2019 
Keywords
 35B40
 35J10
 35J61
 58C15 (primary)
ASJC Scopus subject areas
 Mathematics(all)
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Dive into the research topics of 'Concentration at submanifolds for an elliptic Dirichlet problem near high critical exponents'. Together they form a unique fingerprint.Projects
 1 Finished

Concentration phenomena in nonlinear analysis
Engineering and Physical Sciences Research Council
27/04/20 → 31/03/24
Project: Research council