Existence and multiplicity of solutions to discrete fractional logarithmic Kirchhoff equations

TL;DR

This paper investigates the existence and multiplicity of solutions for a discrete fractional logarithmic Kirchhoff equation, employing variational methods to establish ground state and sign-changing solutions under certain conditions.

Contribution

It introduces new results on the existence and multiplicity of solutions for a discrete fractional Kirchhoff equation with logarithmic nonlinearity, using variational techniques.

Findings

Existence of ground state solutions for p>4.

Existence of sign-changing solutions for p>6.

Multiplicity of nontrivial solutions established.

Abstract

In this paper, we study the discrete fractional logarithmic Kirchhoff equation $$ \left(a+b \int_{\mathbb{Z}^d}|\nabla^s u|^{2} d \mu\right) (-\Delta)^s u+h(x) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb{Z}^d, $$ where $a,\,b>0$ and $0<s<1$. Under suitable assumptions on $h(x)$, we first prove the existence of ground state solutions by the mountain-pass theorem for $p>4$; then we verify the existence of ground state sign-changing solutions based on the method of Nehari manifold for $p>6$. Finally, we establish the multiplicity of nontrivial weak solutions.