Existence And Convergence Of The Least Energy Sign-changing Solutions For Nonlinear Kirchhoff Equations On Locally Finite Graphs

In this paper,we study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation on a locally finite graph G=(V,E),where a,b are positive constants.In the general Nehari manifold method,we can guarantee that if u∈N,u±∈N.However,due to the interference of nonlocal term(∫v |▽u|2dμ)Δu.we are no longer able to obtain the above results,which has caused great problems in our research work.Therefore,we consider If c(x)and f satisfy certain assumptions,in Chapter 2,we will give the relevant embedding lemma on graphs and results on the existence of ground state solutions.Some basic properties on M and related lemmas are illustrated in Chapter 3.Thus in the proof of Chapter 4,we use the constrained variational method to prove the existence of a least energy sign-changing solution ub of the above equation,and to show the energy of ub is strictly larger than twice that of the least energy solutions.Finally,we also study the relationship of the nonlocal Kirchhoff equations with the local equations.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).