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

Shengbing Deng, Fethi Mahmoudi, Monica Musso

OAI: oai:purehost.bath.ac.uk:openaire_cris_publications/4ba77b4c-e9df-4814-9773-53f3106b880a • DOI: https://doi.org/10.1112/plms.12183

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 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.

35B40 35J10 35J61 58C15 (primary) /dk/atira/pure/subjectarea/asjc/2600/2600 name=General Mathematics