Existence and convergence of the least energy sign-changing solutions for nonlinear Kirchhoff equations on locally finite graphs

Abstract

In this paper, we study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation $$\begin{array}{l}\hfill -\mathrm{(}\mathit{a}+\mathit{b}{{\displaystyle \int }}_{\mathbb{V}}|\nabla \mathit{u}{|}^2\mathit{d}\mathit{\mu }\mathrm{)}\mathrm{\Delta }\mathit{u}+\mathit{c}\mathrm{(}\mathit{x}\mathrm{)}\mathit{u}=\mathit{f}\mathrm{(}\mathit{u}\mathrm{)}\end{array}$$ on a locally finite graph $\mathit{G}=\mathrm{(}\mathbb{V}\mathrm{,}\mathbb{E}\mathrm{)}$ , where a, b are positive constants. We use the constrained variational method to prove the existence of a least energy sign-changing solution ${\mathit{u}}_{\mathit{b}}$ of the above equation if $\mathit{c}\mathrm{(}\mathit{x}\mathrm{)}$ and f satisfy certain assumptions, and to show the energy of ${\mathit{u}}_{\mathit{b}}$ is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as $\mathit{b}\to 0^{+}$ , the solution ${\mathit{u}}_{\mathit{b}}$ converges to a least energy sign-changing solution of a local equation $-\mathit{a}\mathrm{\Delta }\mathit{u}+\mathit{c}\mathrm{(}\mathit{x}\mathrm{)}\mathit{u}=\mathit{f}\mathrm{(}\mathit{u}\mathrm{)}$ .