Boundary-layers for a Neumann problem at higher critical exponents

We consider the Neumann problem $(P)\qquad - Δv + v= v^{q-1} \ \text{in }\ \mathcal{D}, \ v > 0 \ \text{in } \ \mathcal{D},\ \partial_νv = 0 \ \text{on } \partial\mathcal{D} ,$ where $\mathcal{D}$ is an open bounded domain in $\mathbb{R}^N,$ $ν$ 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 ${2(N-h)\over N-h-2}.$

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