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

Abstract

Using the Mountain Pass Theorem we show that the problem \begin{equation*} \begin{cases} \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[a,b]\\ u(a)=u(b)=0 \end{cases} \end{equation*} has a solution in anisotropic Orlicz-Sobolev space. We consider Lagrangian $\mathcal{L}=F(t,x,v)+V(t,x)+\langle f(t), x\rangle$ with growth condition determined by anisotropic G-function and some geometric condition of Ambrosetti-Rabinowitz type.