Periodic solutions of Euler-Lagrange equations in an anisotropic Orlicz-Sobolev space setting

TL;DR

This paper establishes the existence of periodic solutions for Euler-Lagrange equations involving anisotropic operators within Orlicz-Sobolev spaces, broadening the theoretical framework for such differential equations.

Contribution

It introduces a unified variational approach in anisotropic Orlicz-Sobolev spaces to find periodic solutions of complex Euler-Lagrange equations, including p-Laplace and (p,q)-Laplace types.

Findings

Existence of periodic solutions proven using variational methods.

Unified framework applicable to a broad class of anisotropic differential operators.

Extension of solution theory to more general operators in Orlicz-Sobolev spaces.

Abstract

In this paper we consider the problem of finding periodic solutions of certain Euler-Lagrange equations, which include, among others, equations involving the $p$-Laplace and, more generality, the $(p,q)$-Laplace operator. We employ the direct method of the calculus of variations in the framework of anisotropic Orlicz-Sobolev spaces. These spaces appear to be useful in formulating a unified theory of existence for the type of problem considered.