Asymptotic behavior of a fourth order mean field equation with Dirichlet boundary condition

TL;DR

This paper analyzes the asymptotic behavior of solutions to a fourth order mean field equation with Dirichlet boundary conditions in a bounded domain, providing a complete characterization of unbounded solutions under bounded parameter conditions.

Contribution

It offers a comprehensive analysis of the asymptotic behavior of unbounded solutions to a specific fourth order mean field equation in four dimensions.

Findings

Complete characterization of unbounded solutions' asymptotics.

Conditions under which solutions become unbounded.

Insights into the behavior of solutions as parameters vary.

Abstract

We consider asymptotic behavior of the following fourth order equation \[ \Delta^2 u= \rho \frac{e^{u}}{\int_\Om e^{u} dx} {in} \Om, u= \partial_\nu u=0 {on} \partial \Omega \] where $\Om$ is a smooth oriented bounded domain in $\R^4$. Assuming that $0<\rho \leq C$, we completely characterize the asymptotic behavior of the unbounded solutions.