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

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 $\mathcal{L}=F(t,x,v)+V(t,x)+⟨f(t),x⟩$, 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.

15 pages, 3 figures

References in corpus (1)

• Mountain pass solutions to Euler-Lagrange equations with general anisotropic operator