Nodal solutions to Paneitz-type equations

On a closed Riemannian manifold (Mⁿ ,g) with a proper isoparametric function f we consider the equation Δ² u -α Δ u +β u = uq, where α and β are positive constants satisfying that α² ≥ 4 β. We let m be the minimum of the dimensions of the focal varieties of f and qf = n- m+4n- m-4, qf = ∞ if n≤ m+4. We prove the existence of infinitely many nodal solutions of the equation assuming that 1<q<qf. The solutions are f-invariant. To obtain the result, first we prove a C⁰-estimate for positive f-invariant solutions of the equation. Then we prove the existence of mountain pass solutions with arbitrarily large energy.