Minimax Theorems on C1 Manifolds via Ekeland Variational Principle

Let X be a Banach space and Φ : X → R of class C1. We are interested in finding critical points for the restriction of Φ to the manifold M = {u ∈ X : G(u) = 1}, where G : X → R is a C1 function having 1 as a regular value. A point u ∈M is a critical point of the restriction of Φ to M if and only if dΦ(u)|TuM = 0 (see the definition in Section 2). Our purpose is to prove two general minimax principles to find almost critical points ofΦ restricted toM. A compactness condition of (PS) type will then imply the existence of a critical point. The applications of minimax principles in the theory of elliptic PDEs are well known and the reader is referred, for instance, to [15] for a thorough introduction to the subject. For applications of minimax principles on C1 manifolds, we refer, for instance, to [2, 8, 11, 13, 16]. In this paper, we present two general minimax principles, Theorems 2.1 and 2.6, for functionals Φ restricted to M. The first one, Theorem 2.1, is a theorem of “mountain-pass type” and the second one, Theorem 2.6, is a theorem of “Ljusternik-Schnirelman type.” A standard approach to prove such results is to first derive a deformation lemma on the manifold M. In the case of Theorem 2.6, one would ask furthermore the deformation to be symmetric, that is, equivariant under the action of the group Z2. Classically the deformation homotopy is constructed with the help of integral lines of a pseudogradient vector field of Φ on M. Since the construction of the integral lines requires the vector field to be locally Lipschitz