We show that the Moser functional J (u) = R (e 4⇡u 2 1) dx on the set B = {u 2 H 1 0 () : kruk 2 1}, where ⇢ R 2 is a bounded domain, fails to be weakly continuous only in the following exceptional case. Defineup 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.
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