ℤ2-equivariant Ljusternik-Schnirelman theory for non-even functionals

Abstract

We develop an equivariant Ljusternik-Schnirelman theory for non-even functionals. We show that if one applies a ℤ2-equivariant min-max procedure to a non-symmetric functional ϕ, then one gets either the usual critical points, defined by ϕ′(x) = 0 or an interesting new class of points x, defined by ϕ (x) = ϕ(−x) and ϕ′(x) = λϕ′(−x) for some λ > 0. We call them “ℤ2-resonant points”; by a “virtual critical point” we understand a point which is either critical or ℤ2-resonant. We extend the classical existence and multiplicity results of Ljusternik-Schnirelman theory for critical points of even functionals to virtual critical points of non-even functionals. As an application we prove a bifurcation-type result for a class of non-homogenous semi-linear elliptic boundary value problems.