Extremising eigenvalues of the GJMS operators in a fixed conformal class

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$. If $s$ is a positive integer satisfying $2s<n$, we let $P_g^s$ be the GJMS operator of order $2s$ in $M$. We investigate in this paper the extremal values taken by fixed eigenvalues of $P_h^s$ as $h$ runs though the whole conformal class $[g]$. Due to the conformal covariance of $P_g^s$ we need only consider two variational problems: maximising negative eigenvalues of $P_h^s$ and minimising positive eigenvalues of $P_h^s$ when $h \in [g]$. These extremal values are conformal invariants of $(M,g)$ and optimisers for these problems, when they exist, are known to not be smooth metrics in general. To overcome this we define and investigate eigenvalues for singular conformal metrics, that we call \emph{generalised eigenvalues}. We develop a new variational framework for renormalised eigenvalues of any index over the set of admissible (singular) conformal factors: we obtain semi-continuity results and Euler-Lagrange equations for local extremals. Using this framework we prove, under mild assumptions on $(M,g)$ and $s$, several new (non)-existence results for extremals of renormalised eigenvalues over $[g]$. These include, among other results, a maximisation result for negative eigenvalues, the minimisation of the principal eigenvalue of $P_g^s$ and the analysis of several invariants of the round sphere $(\mathbb{S}^n, g_0)$. We also establish a strong connection between the existence of optimisers and (nodal) solutions of prescribed $Q$-curvature equations. Our analysis covers the case of any order $s \ge 1$ and allows $P_g^s$ to have kernel. When $s=1$ and $P_g^1$ is the conformal Laplacian, this problem was previously investigated when $k=1, 2$. Our work strongly generalises previous results to eigenvalues of any index.