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 , where the case k = 1 is considered.
- 58C15 (primary)
ASJC Scopus subject areas
Deng, S., Mahmoudi, F., & Musso, M. (2019). Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents. Proceedings of the London Mathematical Society, 118(2), 379-415. https://doi.org/10.1112/plms.12183