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

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