Single Blow up Solutions for a Slightly Subcritical Biharmonic Equation

Abstract

In this paper, we consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent $(P_\epsilon): \Delta^2u=u^{9-\epsilon}, u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\R^5$ and $\epsilon >0$. We study the asymptotic behavior of solutions of $(P_\epsilon)$ which are minimizing for the Sobolev qutient as $\epsilon$ goes to zero. We show that such solutions concentrate around a point $x_0\in\Omega$ as $\epsilon\to 0$, moreover $x_0$ is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point $x_0$ of the Robin's function, there exist solutions concentrating around $x_0$ as $\epsilon$ goes to zero.