Concentration on minimal submanifolds for a singularly perturbed Neumann problem

Abstract

We consider the equation $- \e^2 \D u + u= u^p$ in $\Omega \subseteq {\mathbb{R}}^N$, where $\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\partial Ø$, for $N\ge 3$ 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 $k=1$.

62 pages. To appear in Adv. in Math