Positive solutions for a fourth order equation invariant under isometries

Abstract:

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 5$. We consider the problem \begin{equation*} \tag {$\star $}\Delta _g^2 u+\alpha \Delta _g u+au=f u^{\frac {n+4}{n-4}}, \end{equation*} where $\Delta _g=-div_g(\nabla )$, $\alpha$, $a\in \mathbb {R}$, $u$, $f\in C^{\infty }(M)$. We require $u$ to be positive and invariant under isometries. We prove existence results for $(\star )$ on arbitrary compact manifolds. This includes the case of the geometric Paneitz-Branson operator on the sphere.