Existence and convergence of ground state solutions for Choquard-type systems on lattice graphs

Abstract

In this paper, we study the $p$-Laplacian system with Choquard-type nonlinearity $$ \begin{cases}-\Delta_{p} u+(\lambda a+1)|u|^{p-2} u=\frac{1}{\gamma} \left(R_\alpha\ast F(u,v)\right)F_{u}(u, v), \\ -\Delta_{p} v+(\lambda b+1)|v|^{p-2} v=\frac{1}{\gamma} \left(R_\alpha\ast F(u,v)\right)F_{v}(u, v),\end{cases} $$ on lattice graphs $\mathbb{Z}^N$, where $\alpha \in(0,N),\,p\geq 2,\,\gamma> \frac{(N+\alpha)p}{2N},\,\lambda>0$ is a parameter and $R_{\alpha}$ is the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under some assumptions on the functions $a,\,b$ and $F$, we prove the existence and asymptotic behavior of ground state solutions by the method of Nehari manifold.