Existence results for a generalized mean field equation on a closed Riemann surface

Abstract

Let $\hat{I}\pounds$ be a closed Riemann surface, $h$ a positive smooth function on $\hat{I}\pounds$, $\ddot{I}$ and $\hat{I}\pm$ real numbers. In this paper, we study a generalized mean field equation \begin{align*} -Δu=ρ\left(\dfrac{he^u}{\int_Σhe^u}-\dfrac{1}{\mathrm{Area}\left(Σ\right)}\right)+α\left(u-\fint_Σu\right), \end{align*} where $\hat{I}”$ denotes the Laplace-Beltrami operator. We first derive a uniform bound for solutions when $\ddot{I}\in (8k\ddot{I}€,8(k+1)\ddot{I}€)$ for some non-negative integer number $k\in \mathbb{N}$ and $\hat{I}\pm \notin \mathrm{Spec}\left(-\hat{I}”\right)\setminus \{0\}$. Then we obtain existence results for $\hat{I}\pm <\hat{I}{»}_1\left(\hat{I}\pounds \right)$ by using the Leray-Schauder degree theory and the minimax method, where $\hat{I}{»}_1\left(\hat{I}\pounds \right)$ is the first positive eigenvalue for $-\hat{I}”$.