Ground states of a Kirchhoff equation with the potential on the lattice graphs

Abstract

<abstract><p>In this paper, we study the nonlinear Kirchhoff equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} -\Big(a+b\int_{\mathbb{Z}^{3}}|\nabla u|^{2} d \mu\Big)\Delta u+V(x)u = f(u) \end{align*} $\end{document} </tex-math></disp-formula></p> <p>on lattice graph $ \mathbb{Z}^3 $, where $ a, b > 0 $ are constants and $ V:\mathbb{Z}^{3}\rightarrow \mathbb{R} $ is a positive function. Under a Nehari-type condition and 4-superlinearity condition on $ f $, we use the Nehari method to prove the existence of ground-state solutions to the above equation when $ V $ is coercive. Moreover, we extend the result to noncompact cases in which $ V $ is a periodic function or a bounded potential well.</p></abstract>