Existence of Two Periodic Solutions to General Anisotropic Euler-Lagrange Equations

Abstract

This paper is concerned with the following Euler-Lagrange system \[ \begin{cases} \frac{d}{dt} \mathcal{L}_v(t,u(t),\dot{u}(t)) = \mathcal{L}_x(t,u(t),\dot{u}(t)) \quad \textrm{for a.e. $t \in [-T,T]$}, \\ u(-T) = u(T), \\ \mathcal{L}_v(-T,u(-T),\dot{u}(-T)) = \mathcal{L}_v(T,u(T),\dot{u}(T)), \end{cases} \] 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.