Existence Of Solutions For Fractional Schr(?)dinger-Poisson Systems

In this paper,we study the fractional Schrodinger-Poisson system(?)where 0<s,t?1,(-?)sdenotes the fractional Laplacian of order s.V(x)is a continuous potential and f(x,u)is the nonlinearity.The main results are as follows:In Chapter 2,we study superlinear fractional Schrodinger-Poisson system with positive po-tential.We assume inf V(x)>0 and a condition about measure,so the compactness of Sobolev embedding can be recovered.Because of the addition of Poisson equation,the degree of energy function increases.The absence of A-R condition makes the boundedness of(PS)c sequence not guaranteed.Therefore,we divide E into a finite dimensional subspace and an infinite dimensional subspace.Moreover,we make the nonlinearity term supercubic.Existence of infinitely many high energy solutions has been proved via Fountain theorem.In Chapter 3,we study superlinear fractional Schrodinger-Poisson system with sign-changing potential.If(?)V(x)>-?,then the embedding E?LP is not compact with V(x)sign-changing.We need a transformation V(x)=V(x)+V0,f(x,u)=f(x,u)+V0u,where(?)V(x)>0.By analysing transformed V,f and the geometric properties of I,we prove that the system(FSP)has infinitely many solutions via symmetric Mountain Pass theorem.In Chapter 4,we study sublinear fractional Schrodinger-Poisson system with positive poten-tial.We assume|f(x,u)|?h1(x)|u|k1-1+h2(x)|u|k2-1,where 1<K1,k2<2,hi(x)?L2/2-ki(R3,R+).Afterwards,we can prove I is bounded below and the boundedness of(PS)sequence.Under certain assumptions,it is proved that the system(FSP)has infinitely many nontrival solutions via Genus theorem and Dual Fountain theorem.