Boundary-layers for a Neumann problem at higher critical exponents

TL;DR

This paper investigates solutions to a Neumann boundary value problem that exhibit boundary-layer blow-up along submanifolds as the exponent approaches a higher critical Sobolev value, revealing complex boundary phenomena.

Contribution

It constructs solutions with boundary-layer blow-up along submanifolds for the Neumann problem at higher critical exponents, extending understanding of boundary concentration phenomena.

Findings

Existence of solutions blowing up along submanifolds

Blow-up occurs as the exponent approaches higher critical Sobolev values

Solutions are constructed in suitable domains

Abstract

We consider the Neumann problem $$(P)\qquad - \Delta v + v= v^{q-1} \ \text{in }\ \mathcal{D}, \ v > 0 \ \text{in } \ \mathcal{D},\ \partial_\nu v = 0 \ \text{on } \partial\mathcal{D} ,$$ 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 ${2(N-h)\over N-h-2}.$