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 − ( a + b ∫ V | ∇ u | 2 d μ ) Δ u + 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 u b of the above equation if c ( x ) and f satisfy certain assumptions, and to show the energy of u b is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as b → 0 + , the solution u b converges to a least energy sign-changing solution of a local equation − a Δ u + c ( x ) u = f ( u ).