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

TL;DR

This work proves the existence of least energy solutions for two Kirchhoff-type equations on locally finite graphs with an asymptotically cubic nonlinearity $f(u)=\lambda u+\eta|u|^2u$, a regime not covered by standard growth conditions. Using constrained variational methods on finite domains and infinite graph spaces, the authors define energy functionals $I$ and $J$ and associated Nehari-type constraints, establishing critical points that minimize energy under precise parameter thresholds $|\lambda|<a\lambda_1$ (or $|\lambda|<a\lambda_1^*$) and $\eta>\eta_0$ (or $\eta>\eta_0^*$). They prove positive minimizers exist and provide minimax characterizations for the least energy levels, with explicit lower bounds on solution norms. The results extend previous Euclidean and graph-based analyses by handling nonlinearities that do not satisfy typical $(C_1)$- or $(C_2)$-type conditions and by offering explicit criteria for the existence of least energy solutions on graphs. Altogether, the paper advances variational methods for Kirchhoff-type problems on discrete structures and informs applications in graph-based PDEs and data analysis.

Abstract

We study the existence of least energy solutions for two Kirchhoff equations with the asymptotically cubic nonlinearity $f(u)=λu+η|u|^2u$ on a locally weighted and connected finite graph $G=(V,E)$. Such nonlinearity satisfies neither $\frac{F(u)}{u^4}\to +\infty$ as $|u|\to\infty$, where $F(u)=\int_0^uf(s)ds$, nor $\frac{f(u)}{u}\to 0$ as $u\to 0$. By utilizing the constrained variational method, we prove that there exist $λ_1\ge 0$ and $η_0\ge 0$ ($λ_1^*\ge 0$ and $η_0^*\ge 0$) such that these two equations have at least a least energy solution if $|λ|<aλ_1$ ($|λ|<aλ_1^*$) and $η>η_0$ ($η>η_0^*$).