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For singular mean field equations defined on a compact Riemann surface, weprove the uniqueness of bubbling solutions as far as blowup points are eitherregular points or non-quantized singular sources. In particular the uniquenessresult covers the most general case improving all previous works ofBartolucci-Jevnikar-Lee-Yang \cite{bart-4,bart-4-2} and Wu-Zhang\cite{wu-zhang-ccm}. For example, unlike previous results, we drop theassumption of singular sources being critical points of a suitably definedKirchoff-Routh type functional. Based on refined estimates which allow a majorsimplification of previous proofs, our new argument is robust and flexibleenough to be applied to a wide range of problems requiring a delicate blowupanalysis.

Asymptotic Analysis and Uniqueness of blowup solutions of non-quantized singular mean field equations