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

Article type: Research Article

Authors: Pan, Guofu | Ji, Chao^{; *}

Affiliations: School of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: In this paper, we study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation −(a+b∫V|∇u|2dμ)Δu+c(x)u=f(u) on a locally finite graph G=(V,E), where a, b are positive constants. We use the constrained variational method to prove the existence of a least energy sign-changing solution ub of the above equation if c(x) and f satisfy certain assumptions, and to show the energy of ub is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as b→0+, the solution ub converges to a least energy sign-changing solution of a local equation −aΔu+c(x)u=f(u).

Keywords: Kirchhoff equations, locally finite graphs, sign-changing solutions, variational methods

DOI: 10.3233/ASY-221819

Journal: Asymptotic Analysis, vol. 133, no. 4, pp. 463-482, 2023