Mountain pass type periodic solutions for Euler-Lagrange equations in anisotropic Orlicz-Sobolev space

TL;DR

This paper proves the existence of periodic solutions for Euler-Lagrange equations in anisotropic Orlicz-Sobolev spaces using the Mountain Pass Theorem, considering different growth conditions of the G-function.

Contribution

It extends the application of the Mountain Pass Theorem to anisotropic Orlicz-Sobolev spaces for Euler-Lagrange equations with specific growth conditions.

Findings

Existence of periodic solutions established.

Conditions on G-function growth near zero.

Applicability to anisotropic Orlicz-Sobolev spaces.

Abstract

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 $\Delta_2\cap\nabla_2$ in infinity and globally. We give conditions on the growth of the potential near zero for both situations.