On a Riemannian manifold of dimension $n \geq 3$, for every positive integer $s$, there exists a conformal covariant differential operator of even order $2s \leq n$, whose leading term is the $s$-th power of the Laplacian (the GJMS operator). For $k = 1$, this is the famous conformal Laplacian that appears in the Yamabe problem. We consider the more general problem of minimizing (resp. maximizing) the positive eigenvalues (resp. negative eigenvalues) of these operators among all metrics with fixed volume in a given conformal class, in the case $2s < n$. In particular, we calculate optimal bounds and provide examples where they are attained, as well as others where they are not, depending on the choice of the manifold, $s$, and the index of the eigenvalue.
