Existence Of Solutions For Several Kinds Of Nonhomogeneous Fractional Schr?dinger Equations And Related Problems

Fractional operators of elliptic type arise in a quite natural way in many different problems.For example,（?）is a kind of fractional operator in the theory of relativistic quantum mechanics,where m>0 denotes the mass of the particle.In this thesis,we investigate the existence of solutions for several kinds of nonhomogeneous fractional Schrodinger equations and Dirac systems by variational methods.This thesis is divided into three parts.In the first part,we first consider the following sublinear fractional Schrodinger equation when a（x）allows sign changing（-Δ）^su+V（x）u=a（x）|u|^q-1u+f（x）,x ∈R^N,where 0<q<1.By mountain pass theorem and the minimization method,we prove that the equation has at least two nontrivial solutions.Next,we consider the following fractional Schrodinger equation when a（x）is always negative（-Δ）^su+V（x）|u|^μ-1u=a（x）|u|^q-1u+f（x）,x ∈R^N,where 1 ≤μ<2_s^*/2,μ/（2_s^*-μ）<q<1.By mountain pass theorem and the minimization method,we prove that the equation has at least two nontrivial solutions.The second part investigates the following fractional Schrodinger-Poisson system （?） where A is a parameter.By the method of perturbation and Moser iterative method,we obtain the existence of one nontrivial solution for the system when λ is large enough.In the third part,we consider the following asymptotically periodic fractional Schrodinger equation （-Δ）^su+V（x）u=f（x,u）,x ∈R^N and obtain the existence of nontrivial solutions when V（x） is positive or indefinite.Finally,in the forth part,we consider the following Dirac system with a sublinear pertubation （?） and obtain its existence of periodic solutions.