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.

We consider a conservative second order Hamiltonian system¨ + ∇V ( ) = 0 in R 3 with a potential V having a global maximum at the origin and a line ∩ {0} = ∅ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system. MSC:34C37, 58F05, 70H05

In this paper we introduce a new homotopy invariant -the cohomological span of LS-Conley index. We prove the theorems on the existence of critical points for a class of strongly indefinite functionals with the gradient of the form Lx + K(x), where L is bounded linear and K is completely continuous. We give examples of Hamiltonian systems for which our methods give better results than the Morse inequalities. We also give a formula for the LS-index of an isolated critical point, which is an extension of the classical Dancer theorem for the case of LS-index.