English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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): ∫Σ udv g = 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.

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