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Some existence results on periodic solutions of Euler-Lagrange equations in an Orlicz-Sobolev space setting. (English) Zbl 1335.49022

Summary: 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 a certain functional \(I\). We give conditions which ensure that \(I\) is finitely defined and differentiable on certain subsets of Orlicz-Sobolev spaces \(W^1 L^\Phi\) associated to an \(N\)-function \(\Phi\). We show that, in some sense, it is necessary for the coercivity that the complementary function of \(\Phi\) satisfies the \(\Delta_2\)-condition. We conclude by discussing conditions for the existence of minima of \(I\).

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35B10 Periodic solutions to PDEs
35A15 Variational methods applied to PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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