On solution sequences of the singular Liouville equations with an integral constraint

In this paper, we analyze the blow-up behavior of sequences {u k } satisfying the following conditions ..uk=|x|2.kVkeukin., ((0.1)) where . .. .2, V k . V in C 1, |.V k | . A, 0 < a . V k . b, 0 . . k . . . (0, .), and ..|x|2.keukdx..1. ((0.2)) Furthermore, we assume that there exists some q . (1, 2) such that rq.2.Br(p)|.uk|qdx..2 ((0.3)) for any B r (p) . .. As a result, we give a new proof of the concentration-compactness theorem for the mean field equation.