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 $(P_{\epsilon }):{\Delta }^{2}u=u^{9-\epsilon },u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^5$ 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.

19 pages