Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface

Let (Σ, g) be a compact Riemannian surface without boundary and λ₁(Σ) be the first eigenvalue of the Laplace-Beltrami operator Δ _g . Let h be a positive smooth function on Σ. Define a functional \({J_{\alpha ,\beta }}\left( u \right) = \frac{1}{2}\int_\sum {\left( {{{\left| {{\nabla _g}u} \right|}^2} - \alpha {u^2}} \right)} d{v_g} - \beta \log \int_\sum {h{e^u}} d{v_g}\) on a function space H = {u ∈ W^1,2(Σ): ∫_Σ ^udv g = 0}. If α < λ₁(Σ) and J_α;8π has no minimizer on H, then we calculate the infimum of J_α;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(Σ), then infu2H J_α;8π(u) = ?∞. If β > 8π, then for any α ∈ R, there holds inf_u∈HJ_α,β(u) = ?∞. Moreover, we consider the same problem in the case that α is large, where higher order eigenvalues are involved.