Some existence results for a Paneitz type problem via the theory of critical points at infinity

In this paper a fourth order equation involving critical growth is considered under the Navier boundary condition: ${\mathrm{\Delta }}^{2}u=K{u}^p$, $u>0$ in Ω, $u=\mathrm{\Delta }u=0$ on ∂Ω, where K is a positive function, Ω is a bounded smooth domain in ${\mathbb{R}}^n$, $n⩾5$ and $p+1=2n/(n-4)$, is the critical Sobolev exponent. We give some topological conditions on K to ensure the existence of solution. Our methods involve the study of the critical points at infinity and their contribution to the topology of the level sets of the associated Euler–Lagrange functional.