Boundary-layers for a Neumann problem at higher critical exponents

Abstract

We consider the Neumann problem $$(P)-\Delta v+v=v^{q-1}\text{in }\mathcal{D},v>0\text{in }\mathcal{D},{\partial }_{\nu }v=0\text{on }\partial \mathcal{D},$$ where $\mathcal{D}$ is an open bounded domain in ${\mathbb{R}}^N,$ $\nu$ is the unit inner normal at the boundary and $q>2.$ For any integer, $1\le h\le N-3,$ we show that, in some suitable domains $\mathcal{D},$ problem $(P)$ has a solution which blows-up along a $h-$dimensional minimal submanifold of the boundary $\partial \mathcal{D}$ as $q$ approaches from either below or above the higher critical Sobolev exponent $\frac{2(N-h)}{N-h-2}.$

13 pages