Existence and Convergence of Solutions for Nonlinear Elliptic Systems on Graphs

Abstract

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

Keywords

Nonlinear elliptic system / Locally finite graph / Ground state solution