Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

Given a domain $\Omega$ of $\mathbb{R}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\pa \Omega$ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial \Omega$ perpendicularly.