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 through the whole conformal class $[g]$. These extremal values -- that we call throughout the paper \emph{conformal eigenvalues} -- are conformal invariants of $(M,g)$ and optimisers for these problems, when they exist, are known to not be smooth metrics in general. In this paper we develop a general framework that allows us to address the the existence theory for extremals of conformal eigenvalues. 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 the conformal eigenvalues 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 allows any order $s \ge 1$ and allows $P_g^s$ to have kernel. Previous results only covered the cases $s=1,2$ and $k=1,2$. Our work strongly generalises these results to any $s \ge 1$ and to eigenvalues of any order.