Dirichlet systems with discrete relativistic operator

Abstract

We are concerned with Dirichlet systems involving the relativistic discrete operator $$ u \mapsto \Delta \left [ \frac{\Delta u(n-1)}{\sqrt{1 - |\Delta u(n-1)|^2}} \right ] \qquad \left (n \in \{1, \ldots, T\} \right ).$$ Here, for $u:\{0, \ldots, T+1\}\to \mathbb{R}^N,$ we denote $\Delta u(n-1):=u(n)-u(n-1)$. Besides an "universal" existence result for a system with a general nonlinearity, we obtain multiplicity of solutions for systems with parameterized nonlinearities. Our approaches mainly rely on Brouwer degree arguments and critical point theory for convex, lower semicontinuous perturbations of $C^1$-functionals.