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

Abstract

We are concerned with solvability of nonlinear systems involving a discrete singular $ϕ$-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 $γ:\mathbb{R}^N\times\mathbb{R}^N\to2^{\mathbb{R}^N\times\mathbb{R}^N}$ is a maximal monotone operator with $0_{\mathbb{R}^N \times \mathbb{R}^N}\in γ(0_{\mathbb{R}^N \times \mathbb{R}^N})$. The mapping $ϕ$ 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 $Δ$ 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 $γ$ 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.