In this paper we study some properties of anisotropic Orlicz and anisotropic Orlicz-Sobolev spaces of vector valued functions for a special class of G-functions. We introduce a variational setting for a class of Lagrangian Systems. We give conditions which ensure that the principal part of variational functional is finitely defined and continuously differentiable on Orlicz-Sobolev space.2010 Mathematics Subject Classification. 46B10 , 46E30 , 46E40.
Using the Mountain Pass Theorem, we establish the existence of periodic solution for Euler-Lagrange equation. Lagrangian consists of kinetic part (an anisotropic G-function), potential part K − W and a forcing term. We consider two situations: G satisfying ∆2∩∇2 in infinity and globally. We give conditions on the growth of the potential near zero for both situations.2010 Mathematics Subject Classification. 46E30 , 46E40.
Using the Mountain Pass Theorem we show that the problemx with growth condition determined by anisotropic G-function and some geometric condition of Ambrosetti-Rabinowitz type.2010 Mathematics Subject Classification. 46E30 , 46E40.
Lagrangian is given by L = F (t, x, v) + V (t, x) + f (t), x , growth conditions are determined by an anisotropic G-function and some geometric conditions at infinity. We consider two cases: with and without forcing term f . Using a general version of the mountain pass theorem and Ekeland's variational principle we prove the existence of at least two nontrivial periodic solutions in an anisotropic Orlicz-Sobolev space.
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