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

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.