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

Abstract

We study the existence of least energy solutions for two Kirchhoff equations with the asymptotically cubic nonlinearity $f\left(u\right)=\lambda u+\eta {\left|u\right|}^{2}u$ on a weighted, connected, and locally finite graph $G=(V,E)$. Such nonlinearity satisfies neither $\frac{F\left(u\right)}{u^4}\to +\infty$ as $\left|u\right|\to \infty$, where $F\left(u\right)={\int }_0^{u}f\left(s\right)ds$, nor $\frac{f\left(u\right)}{u}\to 0$ 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 one least energy solution if $\left|\lambda \right|<a{\lambda }_1$ ($\left|\lambda \right|<a{\lambda }_1^{\ast }$) and $\eta >{\eta }_0$ ($\eta >{\eta }_0^{\ast }$).