A deformation lemma with an application to a mean field equation

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Weighted barycentric sets and singular Liouville equations on compact surfaces (2011, background): “Lucia in [32] obtained an alternative deformation lemma yielding existence of an approximating sequence (wn) of critical points of Jρn for some ρn → ρ.”
An Improved Geometric Inequality via Vanishing Moments, with Applications to Singular Liouville Equations (2012, methodology): “To check that Kλ,ρ is a critical level is rather standard, as one can use a monotonicity method from [49], [63].”
Supercritical conformal metrics on surfaces with conical singularities (2011, background): “This corollary, combined with the arguments in [33] (proved for the regular case, but adapting in a straightforward way to the singular one) allows to prove the next alternative.”