Boundary-layers for a Neumann problem at higher critical exponents

Abstract

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