Solutions to discrete nonlinear Kirchhoff-Choquard equations with power nonlinearity

TL;DR

This paper investigates solutions to a discrete nonlinear Kirchhoff-Choquard equation with power nonlinearity on the lattice, establishing existence results for ground state and sign-changing solutions using variational methods.

Contribution

It introduces new existence results for ground state and sign-changing solutions to a discrete Kirchhoff-Choquard equation with power nonlinearities.

Findings

Existence of ground state solutions for p>2.

Existence of sign-changing solutions for p>4.

Application of variational methods on Nehari manifolds.

Abstract

In this paper, we study the following Kirchhoff-Choquard equation $$ -\left(a+b \int_{\mathbb{Z}^3}|\nabla u|^{2} d \mu\right) \Delta u+h(x) u=\left(R_{\alpha}\ast|u|^{p}\right)|u|^{p-2}u,\quad x\in \mathbb{Z}^3, $$ where $a,\,b>0$, $\alpha \in(0,3)$ are constants and $R_{\alpha}$ 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.