Let （∑, g） be a compact Riemannian surface without boundary and λ1（∑） be the first eigenvalue of the Laplace-Beltrami operator △g. Let h be a positive smooth function on ∑. Define a functional （Jα,β（u）=1/2∫∑（｜ gu｜2）dvg-βlog∫heudvg）on a function spaceH={u∈W1,2（∈）：∫∑UDVG=0}.Ifα〈λ1（∑）and Jα,8π has no minimizer on H, then we calculate the infimum of Jα,8π on H by using the method of blow-up analysis. As a consequence, we give a sufficient condition under which a Kazdan-Warner equation has a solution. If α≥ λ1（E）, then infu∈Hα,8π（u）=-∞.Ifβ〉8π ,then for any α ∈ R, there holds infu∈HJα,8π（u）=-∞. Moreover, we consider the same problem in the case that α is large, where higher order eigenvalues are involved.
