Research On The Existence And Asymptotic Behavior Of Solutions For Some Fractional Schrodinger-Kirchhoff Type Equations

With the development of physics and biology,it is more and more necessary to study the differential equation with fractional Laplacian operator.In this paper,we study the existence,multiplicity,nonexistence and asymptotic behavior of solutions for fractional Schr(?)dinger-Kirchhoff type equations by using variational method and critical point theory.In chapter 1,we mainly introduce the research background,overseas and domestic research status of fractional Schr(?)dinger-Kirchhoff type equations,some preliminary knowledge of the variational method and the main research contents of this paper.In chapter 2,we study a class of Schr(?)dinger-Kirchhoff type equations involving the fractional p-Laplacian and a concave-convex term.Under some appropriate assumptions on the potential function,we obtain the existence of positive solutions and infinitely many solutions of the equations via Mountain Pass theorem,Fountain theorem and Clark theorem.In chapter 3,we study a class of fractional Schr(?)dinger-Kirchhoff type equations involving the steep well potential and a concave-convex term.By using the Mountain Pass theorem,truncation technique,a variant version of the Ekeland’s variational principle and discussing several parameters,we obtain the existence of positive energy solutions and negative energy solutions,and give the asymptotic behavior of positive energy solutions and nonexistence of nontrivial solutions.In chapter 4,we study a class of fractional Schr(?)dinger-Kirchhoff type equations involving steep well potential.With some assumptions about the potential function,we obtain the existence of positive ground state solution and discuss their asymptotic behavior.In chapter 5,we summarize the results of this paper and give directions for future work.