article

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

Authors: Frederic Robert and Juncheng Wei

Issue: Volume 57 (2008), Issue 5, 2039-2060

Abstract:

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