Mountain pass theorem with infinite discrete symmetry

Abstract

We extend an equivariant Mountain Pass Theorem, due to Bartsch, Clapp and Puppe for compact Lie groups to the setting of infinite discrete groups satisfying a maximality condition on their finite subgroups. Symmetries play a fundamental role in the analysis of critical points and sets of functionals [2], [20], [12]. The development of Equivariant Algebraic Topology, particularly Equivariant Homotopy Theory, has given a number of tools to conclude the existence of critical points in problems which are invariant under the action of a compact Lie group, as investigated in [11]. In this work we discuss extensions of methods of Equivariant Algebraic Topology to the setting of actions of infinite groups. The main result of this note is the modification of a result by Bartsch, Clapp and Puppe originally proved for actions of compact Lie groups, to infinite discrete groups with appropriate families of finite subgroups inside them. Theorem 1.1 (Mountain Pass Theorem). Let G be an infinite discrete group acting by bounded linear operators on a real Banach space E of infinite dimension. Suppose that G satisfies the maximality condition 1.2 and that the linear action is proper outside 0. Let φ : E → R be a G-invariant functional of class C2−. For any value a ∈ R, define the sublevel set φ = {x ∈ E | φ(x) ≤ a} and the critical set K = ∪c∈RKc, where Kc is the critical set at level c, Kc = {u | ‖φ ′ (u)‖ = 0 φ(u) = c}. Suppose that • φ(0) ≤ a and there exists a linear subspace Ê ⊂ E of finite codimension such that Ê∩φ is the disjoint union of two closed subspaces, one of which is bounded and contains 0. • The functional φ satisfies the Orbitwise Palais-Smale condition 1.3. • The group G satisfies the maximal finite subgroups condition 1.2. Then, the equivariant Lusternik-Schnirelmann category of E relative to φ, G− cat(E, φ) is infinite. If moreover, the critical sets Kc are cocompact under the group action, meaning that the quotient spaces G Kc are compact, then φ(K) is unbounded above. Recall that given a natural number r, the class Cr− denotes the class of functions whose derivatives up to order r exist and are locally Lipschitz. Condition 1.2 restricts maximal finite subgroups and their conjugacy relations. Condition 1.2. Let G be a discrete group and MAX be a subset of finite subgroups. G satisfies the maximality condition if • There exists a prime number p such that every nontrivial finite subgroup is contained in a unique maximal p-group M ∈MAX . • M ∈ MAX =⇒ NG(M) = M , where NG(M) denotes the normalizer of M in G.