The Existence And Concentration Of Solutions For Fractional Schr(?)dinger-Poisson System Via Variational Methods

In this thesis,we study the existence,multiplicity and concentration phenomenon of solutions for non-local elliptic type system,by using the critical point theory and varia-tional analysis of nonlinear analysis.Some new results are obtained.The organization of this thesis is as follows.In chapter 1,the historical background,status and the up-to-date progres for all the investigated problems are introduced,the main contents of the dissertaion are outlined,and some preliminary tools used in the proof of main results are given.In chapter 2,by Nehari manifold and Ljusternik-Schnirelmann theory,we study the following fractional Schr(?)dinger-Poisson system where ?>0 is a small parameter,s,t?(3/4,1).Under a local condition imposed on the potential V,we construct a family of positive ground state solutions which concentrates around a local minimum of V as ? ? 0.In chapter 3,we study the existence and concentration behavior of ground state solutions of the fractional Schr(?)dinger-Poisson system with critical nonlinearity where ?>0 is a small parameter,?>0,4s+2t/s+t<p<2*s =6/3-2s,s,tE(0,1)and satisfies 2t+2s>3.Without the monotonic condition,we obtain a positive ground state solution for ?>0 small,and we show that these ground state solutions concentrate around a local minimum of V as ? ? 0 and decay estimate.In chapter 4,we consider the fractional Schr(?)dinger-Poisson equation where s E(3/4,1),p?(4s+6/3,6/3-2s)and ??R is an undetermined parameter.We deal with the case where the associated functional is not bounded below on the L2-unit sphere.We show the existence of infinitely many solutions(u,?)with u having prescribed L2-norm.