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

Abstract

We are concerned with solvability of nonlinear systems involving a discrete singular $\varphi$-Laplacian operator of type \begin{equation*} u \mapsto Δ\left[ϕ(Δu(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(ϕ(Δu(0)),-ϕ(Δu(T))\right)\inγ(u(0),u(T+1)), \end{equation*} where $\gamma :{\mathbb{R}}^N\times {\mathbb{R}}^N\to 2^{{\mathbb{R}}^N\times {\mathbb{R}}^N}$ 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 $\varphi$ 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.