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

TL;DR

This paper establishes the existence and asymptotic properties of ground state solutions for a class of nonlinear p-Laplacian systems with Choquard-type nonlinearity on lattice graphs, using the Nehari manifold method.

Contribution

It introduces new results on ground state solutions for Choquard-type systems on lattice graphs, extending previous work to discrete fractional Laplacian contexts.

Findings

Proved existence of ground state solutions under specified conditions.

Analyzed the asymptotic behavior of solutions as parameters vary.

Applied Nehari manifold method to a discrete fractional Laplacian system.

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.