Projects per year
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)
Fingerprint
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 Active

Concentration phenomena in nonlinear analysis
Engineering and Physical Sciences Research Council
27/04/20 → 26/04/23
Project: Research council