| The fractional Schrodinger-Poisson-Slater equation is the basic equation for simulating particle motion in random field.These theoretical results related to fractional differential equations are widely used in fractional quantum mechanics,physics and chemistry,obstacle problems,optimization and finance,conformal geometry and minimum surface,etc.In recent years,nonlinear equations involving fractional laplacian operator have attracted great attention because many classical methods cannot be directly applied to fractional cases.The fractional Schrodinger-Poisson-Slater equation is a kind of differential equations with fractional differential operator,the study of its ground state solutions is of great theoretical significance.The main contents of this thesis include the following two parts:In the first part,the existence of ground state solutions for the fractional Schr(?)dinger-Poisson-Slater equation is studied.Since the energy functional of this equation is C~1 smooth,the existence of the ground state solutions can be obtained directly by finding the minimum of the energy functional.Firstly,we prove that the energy function is mandatory on the weighted Sobolev spaceH(R~3)by using embedding inequality and some properties of compact operator.Then,we illustrate that the lower bound of the energy functional is a reachability finite value.Furthermore,it is proved that the critical point of the energy functional exists in the weighted Sobolev spaceH(R~3),and then the existence result of the ground state solutions of the equation is obtained.In the second part,the multiplicity of solutions of the fractional Schr(?)dinger-Poisson-Slater equation is further studied.We first define a minimum value sequence using the related results of part 1,and then show that the energy functional satisfies all the conditions of the symmetrical mountain pass lemma,thus obtaining the multiplicity result of the solutions of the equation.Up to now,there are few studies on the existence of ground state solutions and the multiplicity of solutions for the fractional Schr(?)dinger-Poisson-Slater equation.Therefore,the existence of ground state solutions and the multiplicity of solutions of the fractional Schr(?)dinger-Poisson-Slater equation is mainly studied in this thesis.These theoretical results will help to rich the study of the kind of fractional variational problems. |
