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

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 II. We give conditions which ensure that II is finitely defined and differentiable on certain subsets of Orlicz–Sobolev spaces W1LΦW1LΦ associated to an NN-function ΦΦ. We show that, in some sense, it is necessary for the coercivity that the complementary function of ΦΦ satisfy the Δ2Δ2-condition. We conclude by discussing conditions for the existence of minima of II.