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Turkish Journal of Mathematics

DOI

10.3906/mat-1803-103

Abstract

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

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Last Page

2416

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