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

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

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