Single Blow up Solutions for a Slightly Subcritical Biharmonic Equation

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.Comment: 19 page

Existence of Conformal Metrics on Spheres with Prescribed Paneitz Curvature

In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the $n$-sphere, with $n\geq 5$. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given positive function under which we prove the existene of a metric, conformally equivalent to the standard metric, with prescribed Paneitz curvature.Comment: 20 page

The Paneitz Curvature Problem on Lower Dimensional Spheres

In this paper we prescribe a fourth order conformal invariant 9the Paneitz Curvature) on five and six spheres. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results.Comment: 34 page