Existence of two periodic solutions to general anisotropic Euler-Lagrange equations

TL;DR

This paper proves the existence of at least two nontrivial periodic solutions for a class of anisotropic Euler-Lagrange systems using variational methods, under certain growth and geometric conditions.

Contribution

It introduces new existence results for periodic solutions of anisotropic Euler-Lagrange equations with general growth conditions, extending previous work to more general Lagrangians.

Findings

Existence of two nontrivial periodic solutions proven.

Results hold with and without forcing term.

Solutions found in anisotropic Orlicz-Sobolev space.

Abstract

This paper is concerned with the following Euler-Lagrange system \[ \frac{d}{dt}\mathcal{L}_v(t,u(t),\dot u(t))=\mathcal{L}_x(t,u(t),\dot u(t))\quad \text{ for a.e. }t\in[-T,T],\quad u(-T)=u(T), \] where Lagrangian is given by $\mathcal{L}=F(t,x,v)+V(t,x)+\langle f(t), x\rangle$, 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.