Abstract

We show that the Moser functional J = ∫Ω dx on the set B = {u ϵ H10: ||∆u||2 ≤ 1}, where Ω R2 is a bounded domain, fails to be weakly continuous only in the following exceptional case. Define gsw = s-1/2w for s > 0. If uk → u in B while lim inf J > J, then, with some sk → 0, uk = gsk [-1/2 min {1,log 1/|x|}], up to translations and up to a remainder vanishing in the Sobolev norm. In other words, the weak continuity fails only on translations of concentrating Moser functions. The proof is based on a profile decomposition similar to that of Solimini [16], but with different concentration operators, pertinent to the two-dimensional case.