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

Abstract

International audienceIn 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 W 1 L 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

Clarke duality for Hamiltonian systems with nonstandard growth

“…Next, we will be concerned with the notion of G-function and Orlicz spaces. We refer the reader to [13,17] for more comprehensive information about convex functions and to [2,3,4,18,20] for more information on anisotropic G-functions and Orlicz spaces.…”

“…is an equivalent norm to · W 1 L G (see [2,Remark 1]). We set W 1 T L G := u ∈ W 1 L G : u(0) = u(T ) and…”

“…T 0 v(t) dt = 0 . In the space W 1 L G an anisotropic version of Poincaré-Wirtinger inequality holds (see [2] or [4]):…”

“…Lemma 2.6 (see [2,Corollary 2.5]). If u k is a bounded sequence in Orlicz-Sobolev space then u k has a uniformly convergent subsequence.…”

“…In [2,Theorem 4.5], it was proved that if there exist Λ 0 , λ 0 > 0 and functions a ∈ C(R 2n , R) and b ∈ L 1 ([0, T ], R) such that…”