Existence and Convergence of Ground State Solutions for a ( p,q )-Laplacian System on Weighted Graphs

Abstract

We investigate the existence of ground state solutions for a $(p,q)$ -Laplacian system with $p,q>1$ and potential wells on a weighted locally finite graph $G=(V,E)$ . By making use of the method of Nehari manifold and the Lagrange multiplier rule, we prove that if the nonlinear term $F$ takes on the super- $(p,q)$ -linear growth and the potential functions $a(x)$ and $b(x)$ satisfy some suitable conditions, then for any fixed parameter $\lambda \ge 1$ , the system is provided with a ground state solution $(u_{\lambda },v_{\lambda })$ . Additionally, we set up the convergence property of the solutions set $\{(u_{\lambda },v_{\lambda })\}$ when $\lambda \to +\mathrm{\infty }$ .