Some existence results on periodic solutions of Euler–Lagrange equations in an Orlicz–Sobolev space setting

Abstract

In this paper we consider the problem of finding periodic solutions of certain Euler-Lagrange equations. We employ the direct method of the calculus of variations, i.e. we obtain solutions minimizing certain functional I. We give conditions which ensure that I is finitely defined and differentiable on certain subsets of Orlicz-Sobolev spaces W1L Φ associated to an N-function Φ. We show that, in some sense, it is necessary for the coercitivity that the complementary function of Φ satisfy the ∆2-condition. We conclude by discussing conditions for the existence of minima of I.