Systems with discrete singular $\phi$-Laplacian and maximal monotone boundary conditions

TL;DR

This paper investigates the solvability of nonlinear discrete systems involving a singular $$-Laplacian operator with maximal monotone boundary conditions, using variational methods and a priori estimates.

Contribution

It introduces new existence results for solutions of discrete $$-Laplacian systems with maximal monotone boundary conditions, employing both a priori estimates and variational techniques.

Findings

Existence of solutions via a priori estimates when nonlinearity lacks potential structure.

Variational approach yields solutions as minimizers or saddle points for gradient-type nonlinearities.

Established conditions under which solutions exist for the nonlinear discrete systems.

Abstract

We are concerned with solvability of nonlinear systems involving a discrete singular $\phi$-Laplacian operator of type \begin{equation*} u \mapsto \Delta\left[\phi(\Delta u(n-1))\right] \qquad (n\in \{1, \dots, T\}), \end{equation*} associated with a general two point boundary condition having the form \begin{equation*} \left(\phi(\Delta u(0)),-\phi(\Delta u(T))\right)\in\gamma(u(0),u(T+1)), \end{equation*} where $\gamma:\mathbb{R}^N\times\mathbb{R}^N\to2^{\mathbb{R}^N\times\mathbb{R}^N}$ is a maximal monotone operator with $0_{\mathbb{R}^N \times \mathbb{R}^N}\in \gamma(0_{\mathbb{R}^N \times \mathbb{R}^N})$. The mapping $\phi$ is a potential homeomorphism from an open ball of radius $a$ centered at the origin $B_a \subset \mathbb{R}^N$ onto $\mathbb{R}^N$ and $\Delta$ stands for the usual forward difference operator. When the perturbing nonlinearity in the system has not a potential structure we obtain existence of solutions by a priori estimates. Also, when the nonlinearity is of gradient type and $\gamma$ is a subdifferential, we provide a variational approach of the system in the frame of critical point theory for convex, lower semicontinuous perturbations of $C^1$-functionals. Then we derive the existence of solutions either as minimizers or saddle points of the corresponding energy functional.