  We consider a singularly perturbed problem with mixed Dirichlet and Neumann
boundary conditions in a bounded domain $\Omega\subset\R^{n}$ whose boundary
has an $(n-2)$-dimensional singularity. Assuming $1<p<\frac{n+2}{n-2}$, we
prove that, under suitable geometric conditions on the boundary of the domain,
there exist solutions which approach the intersection of the Neumann and the
Dirichlet parts as the singular perturbation parameter tends to zero.
