Positive solutions for a fourth order equation invariant under isometries

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Let ( M , g ) (M,g) be a smooth compact Riemannian manifold of dimension n ≥ 5 n\geq 5 . We consider the problem ( ⋆ ) Δ g 2 u + α Δ g u + a u = f u n + 4 n − 4 , \begin{equation*} \tag {$\star $}\Delta _g^2 u+\alpha \Delta _g u+au=f u^{\frac {n+4}{n-4}}, \end{equation*} where Δ g = − d i v g ( ∇ ) \Delta _g=-div_g(\nabla ) , α \alpha , a ∈ R a\in \mathbb {R} , u u , f ∈ C ∞ ( M ) f\in C^{\infty }(M) . We require u 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.