Let (Σ, g) be a compact Riemannian surface without boundary and W 1,2 (Σ) be the usual Sobolev space. For any real number p > 1 and α ∈ R , we define a functional Jα,8π(u) = 1 2 (∫ Σ |∇gu| 2 dvg − α(∫ Σ |u| p dvg) 2/p) − 8π log ∫ Σ he u dvg on a function space H = { u ∈ W 1,2 (Σ) : ∫ Σ udvg = 0 } , where h is a positive smooth function on Σ. Denote λ1,p(Σ) = inf u∈H, ∫ Σ |u| p dvg =1 ∫ Σ |∇gu| 2 dvg. If α < λ1,p(Σ) and Jα,8π has no minimizer on H , then we obtain the exact value of infH Jα,8π by using a method of blow-up analysis. Hence, if infH Jα,8π is not equal to that value, then Jα,8π|H has a critical point that satisfies a Kazdan-Warner equation. This recovers a recent result of Yang and Zhu

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