Solutions to discrete nonlinear Kirchhoff-Choquard equations with power nonlinearity

Abstract

In this paper, we study the following Kirchhoff-Choquard equation $$-\left(a+b{\int }_{{\mathbb{Z}}^3}|\nabla u{|}^{2}d\hat{I}¼\right)\hat{I}”u+h(x)u=\left(R_{\hat{I}}\pm \ast |u{|}^p\right)|u{|}^{p-2}u,x\in {\mathbb{Z}}^3,$$ where $a,b>0$, $\hat{I}\pm \in (0,3)$ are constants and $R_{\hat{I}}\pm$ is the Green's function of the discrete fractional Laplacian that behaves as the Riesz potential. Under some suitable assumptions on potential function $h$, for $p>2$, we first establish the existence of ground state solutions based on the Nehari manifold. Subsequently, for $p>4$, we obtain the existence of ground state sign-changing solutions by adopting constrained minimization arguments on the sign-changing Nehari manifold.

18 pages. arXiv admin note: text overlap with arXiv:2404.11856, arXiv:2407.09794