Existence and Convergence of Solutions for Nonlinear Elliptic Systems on Graphs

We consider a kind of nonlinear systems on locally finite graph $$G=(V,E)$$ . prove via the mountain pass theorem that this has nontrivial ground state solution which depends parameter $$\lambda $$ with some suitable assumptions potentials. Moreover, we pay attention to concentration behavior these solutions and as \rightarrow \infty , converge corresponding Dirichlet problem. Finally, also provide numerical experiments illustrate our results.