Concentration on minimal submanifolds for a singularly perturbed Neumann problem

We consider the equation - \e^2 \D u + u= u^p in $\Omega \subseteq \R^N$, where $\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of \partial \O, for $N \geq 3$ and for $k \in \{1, ..., N-2\}$. We impose Neumann boundary conditions, assuming $1<p <\frac{N-k+2}{N-k-2}$ and \e \to 0^+. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.Comment: 62 pages. To appear in Adv. in Mat