# A double mean field equation related to a curvature prescription problem

## TL;DR

This paper investigates a double mean field PDE linked to curvature prescription on compact surfaces with boundary, providing blow-up analysis, inequalities, and existence results for solutions across various parameter ranges.

## Contribution

It introduces new analytical techniques for a double mean field PDE, including blow-up analysis, a Moser-Trudinger inequality, and existence results for min-max solutions.

## Key findings

- Energy-minimizing solutions for certain parameters
- Existence of min-max solutions for a broader parameter range
- Results are dense if the surface is not simply connected

## Abstract

We study a double mean field-type PDE related to a prescribed curvature problem on compacts surfaces with boundary. We provide a general blow-up analysis, then a Moser-Trudinger inequality, which gives energy-minimizing solutions for some range of parameters. Finally, we provide existence of min-max solutions for a wider range of parameters, which is dense in the plane if $\S$ is not simply connected.