In this paper, we study existence of solutions with boundary layer and peaks for an elliptic problem. We prove that the problem has a mountain-pass-type solution, which has a boundary layer and a single peak near the boundary. Moreover, we also study the existence of solutions with interior peak.

We study an elliptic problem involving critical Sobolev exponent in domains with small holes. We prove the existence of solutions which blow up like a volcano near the centre of each hole.