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

Abstract

We consider the equation $-{\epsilon }^2\Delta u+u=u^p$ 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 $\partial \Omega$, 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 $\epsilon \to 0^{+}$. This result settles in full generality a phenomenon previously considered only in the particular case $N=3$ and $k=1$.