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

TL;DR

This paper investigates the existence and bounds of solutions to a generalized mean field equation on closed Riemann surfaces, extending understanding of such equations with new existence results under specific spectral conditions.

Contribution

The paper establishes uniform bounds for solutions within certain parameter ranges and proves existence results using topological and variational methods, considering the spectral properties of the Laplacian.

Findings

Uniform bounds for solutions when re in specific parameter intervals

Existence of solutions for <1() using Leray-Schauder degree and minimax methods

Results depend on spectral properties of the Laplace-Beltrami operator

Abstract

Let $\Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $\Sigma$, $\rho$ and $\alpha$ real numbers. In this paper, we study a generalized mean field equation \begin{align*} -\Delta u=\rho\left(\dfrac{he^u}{\int_\Sigma he^u}-\dfrac{1}{\mathrm{Area}\left(\Sigma\right)}\right)+\alpha\left(u-\fint_{\Sigma}u\right), \end{align*} where $\Delta$ denotes the Laplace-Beltrami operator. We first derive a uniform bound for solutions when $\rho\in (8k\pi, 8(k+1)\pi)$ for some non-negative integer number $k\in \mathbb{N}$ and $\alpha\notin\mathrm{Spec}\left(-\Delta\right)\setminus\set{0}$. Then we obtain existence results for $\alpha<\lambda_1\left(\Sigma\right)$ by using the Leray-Schauder degree theory and the minimax method, where $\lambda_1\left(\Sigma\right)$ is the first positive eigenvalue for $-\Delta$.