Multiple solutions to relativistic systems with potential boundary conditions

Abstract We consider potential systems of differential equations of the form \begin{equation*}-\left[ \phi(u^{\prime}) \right] ^{\prime} = \nabla_u F(t,u),\quad \mbox{in } [0,T],\end{equation*} under the general boundary condition \begin{equation*}\left ( \phi \left( u^{\prime }\right)(0), -\phi \left( u^{\prime }\right)(T)\right )\in \partial j(u(0), u(T)),\end{equation*} where $\phi(y)={y}/{\sqrt{1- |y|^2}}$ and $j:\mathbb{R}^N \times \mathbb{R}^N \rightarrow (-\infty, +\infty]$ is convex and lower semicontinuous. Making use of the variational approach introduced in the recent paper “Potential systems with singular $\phi$ -Laplacian”, we obtain multiplicity of solutions when the action functional is even, as well as existence of multiple geometrically distinct solutions when this functional is invariant with respect to some discrete group.