Multiple critical points for a class of periodic lower semicontinuous functionals

Abstract

We deal with a class of functionals $I$ on a Banach space $X,$having the structure $I=\Psi+\mathcal G,$ with $\Psi : X \to (-\infty , + \infty ]$ proper, convex, lower semicontinuous and$\mathcal G: X \to \mathbb{R} $ of class $C^1.$ Also, $I$ is$G$-invariant with respect to a discrete subgroup $G\subset X$with $\mbox{dim (span}\ G)=N$. Under some appropriate additionalassumptions we prove that $I$ has at least $N+1$ critical orbits.As a consequence, we obtain that the periodically perturbed$N$-dimensional relativistic pendulum equation has at least $N+1$geometrically distinct periodic solutions.