Ground state sign-changing solutions for Kirchhoff-type equations with logarithmic nonlinearity on locally finite graphs

TL;DR

This work studies ground state solutions, including sign-changing ones, for Kirchhoff-type equations with logarithmic nonlinearity on locally finite graphs under Dirichlet boundary on a bounded domain. It develops a direct non-Nehari manifold variational approach, defining two constraint sets that yield a sign-changing ground state with energy $m$ and a positive ground state with energy $c$, and proves the sharp energy relation $m\ge2c$. The analysis navigates the nonlocal Kirchhoff term and graph setting by establishing key decompositions of the energy and a two-parameter maximization framework to control $I(u)$ through $u^+$ and $u^-$. The results extend known Euclidean results to graphs and allow broader nonlinearities, highlighting the potential for graph-based variational methods in nonlinear Kirchhoff problems.

Abstract

We obtain the existence results of ground state sign-changing solutions and ground state solutions for a class of Kirchhoff-type equations with logarithmic nonlinearity on a locally finite graph $G=(V,E)$, and obtain the sign-changing ground state energy is larger than twice of the ground state energy. The method we used is a direct non-Nehari manifold method in [X.H. Tang, B.T. Cheng. J. Differ. Equations. 261(2016), 2384-2402.]

Figures (1)

• Figure 1: The diagram of function $\frac{|s|^{p-2}s\ln |s|^r}{|s|^3}$ with different values