# Concentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem

### Andrea Malchiodi

Scuola Normale Superiore, Pisa, Italy### Fethi Mahmoudi

Universidad de Chile, Santiago, Chile

## Abstract

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