Ground State Sign-changing Solutions For Asymptotically Cubic Nonlinear Kirchhoff Systems With Hartree Term

Kirchhoff equation is a kind of very important nonlinear partial differential equation,and many physical phenomena can be described by it,such as direct-current circuits and alternating-current circuits in physics.Therefore,it is of great theoretical significance and practical value to study such problems.In this thesis,the existence of the ground state sign-changing solutions for Kirchhoff systems with Hartree terms is discussed by means of minimization sequence on sign-changing Nehari set,quantitative deformation lemma and topological degree theory.The thesis is divided into three chapters.In the first chapter,we introduce the preface.This chapter mainly introduces the latest research progress of asymptotically cubic nonlinear Kirchhoff equation,the research content,research ideas and main conclusions of this paper.The second chapter is the preliminary knowledge,which mainly gives the nature of the studied space and several basic lemmas.In the third chapter,we consider the ground state sign-changing solutions for a class of asymptotically cubic nonlinear Kirchhoff systems with Hartree term where a>0,b≥0,the potential function V ∈ C（R³,R₊）,α∈（0,3/2）,p∈（1+α/3,2-α/3）.The innovation of this paper lies in the condition of f.Many scholars require that f has a continuous derivative when studying the sign-changing solution of the equation,but this paper only needs to satisfy the continuity of f and study the case that f is asymptotically cubic nonlinear,that is,（f₁）f∈C（R,R）and lim_t→0 f（t）/t=0;（f₂）the function f（t）/t³ is nonincreasing on（-∞,0）and nondecreasing on（0,∞）respectively,lim_t→-∞ f（t）/t³=β₁ and lim_t→∞f（t）/t³=β₂ with β₁,β₂>0.In this chapter,we first prove that the minimum point can be reached by using the minimization sequence on the sign-changing Nehari set and the weak lower semicontinuity of the norm,and then obtain the following theorems by using the quantitative deformation lemma and topological degree theory.Theorem 3.1 Assume that（V₀）and（f₁）-（f₂）hold.Then the system（3.1）has at least a ground state sign-changing solution.Next,we study the energy relationship between the ground state sign-changing solution and the ground state solution.Through a series of studies,we can get the energy of the ground state sign-changing solution of the system（3.1）is strictly twice that of the ground state solution.