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

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 $N\ge 3$ 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 $k=1$.