@article{Wang2023, 
author = {Jun Wang and Yanni Zhu and Kun Wang},
title = {Existence and asymptotical behavior of the ground state solution for the Choquard equation on lattice graphs},
year = {2023},
journal = {Electronic Research Archive},
volume = {31},
number = {2},
pages = {812-839},
keywords = {ground state solution, Brézis-Lieb Lemma, lattice graph, choquard equation},
url = {https://www.sciopen.com/article/10.3934/era.2023041},
doi = {10.3934/era.2023041},
abstract = {In this paper, we study the nonlinear Choquard equation     −  Δ  u  +  V  (  x  )  u  =      (                  ∑                                            y              ≠              x                                      y              ∈                                                                    Z                                                        N                                                                                                                                      |                        u            (            y            )                                          |                            p                                                          |                        x            −            y                                          |                                            N                −                α                                                          )        |    u            |              p      −      2        uon lattice graph              Z              N      . Under some suitable assumptions, we prove the existence of a ground state solution of the equation on the graph when the function    V is periodic or confining. Moreover, when the potential function    V  (  x  )  =  λ  a  (  x  )  +  1 is confining, we obtain the asymptotic properties of the solution        u    λ   which converges to a solution of a corresponding Dirichlet problem as    λ  →  ∞.}
}