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

TL;DR

This paper proves the existence of least energy solutions for two Kirchhoff equations with asymptotically cubic nonlinearities on finite graphs, using variational methods under specific parameter conditions.

Contribution

It introduces a new approach to establish solutions for Kirchhoff equations with non-standard nonlinearities on graphs, expanding the understanding of such equations in discrete settings.

Findings

Existence of least energy solutions under certain parameter bounds.

Identification of critical parameter thresholds for solutions.

Application of constrained variational methods to graph-based equations.

Abstract

We study the existence of least energy solutions for two Kirchhoff equations with the asymptotically cubic nonlinearity $f(u)=\lambda u+\eta|u|^2u$ 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 $\lambda_1\ge 0$ and $\eta_0\ge 0$ ($\lambda_1^*\ge 0$ and $\eta_0^*\ge 0$) such that these two equations have at least a least energy solution if $|\lambda|<a\lambda_1$ ($|\lambda|<a\lambda_1^*$) and $\eta>\eta_0$ ($\eta>\eta_0^*$).