A threshold phenomenon for embeddings of H0minto Orlicz spaces

Given an open bounded domain Omega subset of R-2m with smooth boundary, we consider a sequence (u(k))(k is an element of N) of positive smooth solutions to {(-Delta)(m)u(k) = lambda(k)u(k)e(mu2/k) in Omega u(k) = partial derivative(nu)u(k) = ... = partial derivative(m-1)(nu)u(k) = 0 on partial derivative Omega, where lambda(k) -> 0(+). Assuming that the sequence is bounded in H-0(m) (Omega), we study its blow- up behavior. We show that if the sequence is not precompact, then lim inf(k ->infinity) parallel to u(k)parallel to(2)(H0m) : = lim inf(k ->infinity) integral(Omega) u(k)(-Delta)(m)u(k)dx >= Lambda(1), where Lambda(1) = (2m - 1)!vol( S-2m) is the total Q-curvature of S-2m.