Abstract
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 ∈ W1,2(Σ): ∫Σudvg = 0}. If α < λ1(Σ) 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(Σ), then infu2H Jα;8π(u) = −∞. If β > 8π, then for any α ∈ R, there holds infu∈HJα,β(u) = −∞. Moreover, we consider the same problem in the case that α is large, where higher order eigenvalues are involved.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11171347, 11471014, 41275063 and 11401575).
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Yang, Y., Zhu, X. Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface. Sci. China Math. 61, 1109–1128 (2018). https://doi.org/10.1007/s11425-017-9086-6
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DOI: https://doi.org/10.1007/s11425-017-9086-6