Existence of solutions to a higher dimensional mean-field equation on manifolds

TL;DR

This paper proves the existence of solutions to a higher-dimensional mean-field equation on closed Riemannian manifolds, extending previous results to more complex geometric settings for specific parameter ranges.

Contribution

It establishes existence results for a class of mean-field equations involving higher-order Laplacians on manifolds, generalizing known cases to higher dimensions and operators.

Findings

Existence of solutions for certain λ values.

Extension of mean-field equation solutions to higher dimensions.

Application to closed Riemannian manifolds.

Abstract

For $m\geq 1$ we prove an existence result for the equation $$(-\Delta_g)^m u+\lambda=\lambda\frac{e^{2mu}}{\int_M e^{2mu}d\mu_g}$$ on a closed Riemannian manifold $(M,g)$ of dimension $2m$ for certain values of $\lambda$.