Nodal solutions to Paneitz-type equations

Abstract

On a closed Riemannian manifold $(M^n,g)$ with a proper isoparametric function $f$ we consider the equation ${\Delta }^{2}u-\alpha \Delta u+\beta u=u^q$, where $\alpha$ and $\beta$ are positive constants satisfying that ${\alpha }^2\ge 4\beta$. We let $\mathbf{m}$ be the minimum of the dimensions of the focal varieties of $f$ and $q_f=\frac{n-\mathbf{m}+4}{n-\mathbf{m}-4}$, $q_f=\infty$ if $n\le \mathbf{m}+4$. We prove the existence of infinitely many nodal solutions of the equation assuming that $1<q<q_f$. The solutions are $f$-invariant. To obtain the result, first we prove a $C^0-$estimate for positive $f$-invariant solutions of the equation. Then we prove the existence of mountain pass solutions with arbitrarily large energy.