Boundary-layers for a Neumann problem at higher critical exponents

We consider the Neumann problem -∇ν + ν = νq-1 in D, ν>0 in D, ∂νv =0 on ∂D, (P) where D is an open bounded domain in RN, ν is the unit inner normal at the boundary and q > 2. For any integer, 1 ≤ h ≤ N - 3, we show that, in some suitable domains D,problem (P) has a solution which blows-up along a h-dimensional minimal submanifold of the boundary ∂D as q approaches from either below or above the higher critical Sobolev exponent 2(N-h)/N-h-2. © 2016 Unione Matematica Italiana.