Multiple solutions to the Bahri-Coron problem in domains with a shrinking hole of positive dimension

Let Omega be a bounded smooth domain in R-N, N >= 3, and let M be a compact smooth submanifold of Omega without boundary such that 1 <= dim M <= N - 2. We consider the problem -Delta u = vertical bar u vertical bar(2*-2)u in Omega(epsilon), u = 0 on partial derivative Omega(epsilon), where Omega(epsilon) : = {x is an element of Omega : dist(x, M) > epsilon}, 2* := 2N/N-2 is the critical Sobolev exponent, and epsilon > 0 is small enough. Under some additional assumptions we obtain the multiplicity of positive and sign changing solutions.