On an eigenvalue problem related to the critical exponent

In this paper we study the eigenvalue problem {-Delta upsilon = N (N - 2)p epsilon lambda u epsilon(p)epsilon(-1)upsilon in Omega upsilon = 0 on partial derivative Omega where Omega subset of R-N is a smooth bounded domain, N >= 3, p epsilon = N+2/N-2 -epsilon,epsilon > 0 and u epsilon is a positive solution of the problem {-Delta u = N(N - 2)u(P)epsilon in Omega {u = 0 on partial derivative Omega such that integral Omega vertical bar triangle u epsilon vertical bar(2)/(integral Omega u epsilon p epsilon+1)2/p epsilon+1 -> S as epsilon -> 0 where S is the best Sobolev constant for the embedding of H-0(1)(Omega) into L-2* (Omega), 2* = 2N/N-2. We prove several estimates for the eigenvalues lambda i,epsilon of (I), i = 2,.., N + 2 and some qualitative properties of the corresponding eigenfunctions.