Least energy solutions of two asymptotically cubic Kirchhoff equations on locally finite graphs

We study the existence of least energy solutions for two Kirchhoff equations with the asymptotically cubic nonlinearity $f(u)=\lambda u+\eta \mathrm{\mid }u{\mathrm{\mid }}^{2}u$ on a locally weighted and connected finite graph $G=(V,E)$. Such nonlinearity satisfies neither $\frac{F(u)}{u^4}\to +\mathrm{\infty }$ as $\mathrm{\mid }u\mathrm{\mid }\to \mathrm{\infty }$, where $F(u)={\int }_0^{u}f(s)ds$, nor $\frac{f(u)}{u}\to 0$ as $u\to 0$. By utilizing the constrained variational method, we prove that there exist ${\lambda }_1\ge 0$ and ${\eta }_0\ge 0$ (${\lambda }_1^{\ast }\ge 0$ and ${\eta }_0^{\ast }\ge 0$) such that these two equations have at least a least energy solution if $\mathrm{\mid }\lambda \mathrm{\mid }<a{\lambda }_1$ ($\mathrm{\mid }\lambda \mathrm{\mid }<a{\lambda }_1^{\ast }$) and $\eta >{\eta }_0$ ($\eta >{\eta }_0^{\ast }$).