On a class of Kazdan--Warner equations

Authors : Yu FANG, Mengjie ZHANG

Pages : 2400-2416

View : 10 | Download : 8

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let $insert ignore into journalissuearticles values(\small{\Si},g);$ be a compact Riemannian surface without boundary and $W^{1,2}insert ignore into journalissuearticles values(\Si);$ be the usual Sobolev space. For any real number $p>1$ and $\alpha\in\mathbb{R}$, we define a functional $$ J_{\alpha,8\pi}insert ignore into journalissuearticles values(u);=\frac{1}{2}\leinsert ignore into journalissuearticles values( \int_\Si |\nabla_g u|^2dv_g-\alpha insert ignore into journalissuearticles values(\int_\Si |u|^pdv_g);^{2/p}\ri);-8\pi\log\int_\Si he^u dv_g $$ on a function space $\mathcal{H}=\le\{u\in W^{1,2}insert ignore into journalissuearticles values(\Si);:\int_{\Si}u dv_{g}=0\ri\}$, where $h$ is a positive smooth function on $\Si$. Denote $$\lambda_{1,p}insert ignore into journalissuearticles values(\Si);=\inf_{u\in \mathcal{H},\,\int_\Si |u|^p dv_g=1}\int_{\Si}|\nabla_{g}u|^{2}\mathrm{d}v_{g}. $$ If $\alpha
Keywords : Trudinger Moser inequality, blow up analysis, Kazdan Warner equation