  We perform a systematic variational method for functionals depending on
eigenvalues of Riemannian manifolds. It is based on a new concept of Palais
Smale sequences that can be constructed thanks to a generalization of classical
min-max methods on $C^1$ functionals to locally-Lipschitz functionals. We prove
convergence results on these Palais-Smale sequences emerging from combinations
of Laplace eigenvalues or combinations of Steklov eigenvalues in dimension 2.
