<p>This paper investigates the existence of generalized solutions to nonlinear inclusion problems involving a competing operator on locally finite graphs. We develop a variational framework for studying differential inclusions with nonsmooth potentials in discrete settings, extending classical results from continuum analysis to graph structures. Using techniques from nonsmooth analysis and Galerkin approximations, we establish existence theorems under general growth conditions. The results have applications in image processing, data analysis, and network modeling, where graph-based operators naturally arise. Our approach handles the competing effects of <i>p</i>-Laplacian and <i>q</i>-Laplacian terms while accommodating nonsmooth nonlinearities through Clarke’s generalized gradient.</p>