  Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. We study
the variational properties of the $k$-th eigenvalue functional $\tilde g\in[g]
\mapsto \lambda_k(L_{\tilde g})$ under a non-volume normalization proposed by
Sire-Xu. We discuss necessary conditions for the existence of extremal
eigenvalues under such normalization. Also, we discuss the general existence
problem when $k=1$.
