The Existence And Multiplicity Of Solutions For Some Coupled Elliptic Systems

In this paper,we study four coupled elliptic systems,namely,a nonhomogeneous coupled elliptic system with Sobolev critical exponent,a coupled Choquard system with Hardy-Littlewood-Sobolev critical exponent,a fractional Schr?dinger system with linear and nonlinear couplings and a fractional Schr?dinger-Poisson system with Hardy poten-tials.Specially,the paper is mainly concerned with the following contents:In Chapter 1,we introduce physical background and current status of the four coupled elliptic systems.In Chapter 2,we introduce fractional Sobolev space,some important inequalities and calculus of variations.In Chapter 3,we study the multiplicity of solution for a nonhomogeneous coupled elliptic system with Sobolev critical exponent over a non-contractible domain,precisely,a smooth bounded annular domain.We first prove the existence of three solutions for the system by splitting Nehari manifold method and Lusternik-Schnirelmann theory,and then apply Mountain Pass Lemma to prove the existence of fourth solution,a high energy solu-tion,when the inner hole of the annulus is sufficiently small.In Chapter 4,we apply linking theory and decomposition of Palais-Smale sequence to prove the existence of high energy solutions for a coupled Choquard system with Hardy-Littlewood-Sobolev critical exponent in R~N.In Chapter 5,we study a fractional Schr?dinger system with linear and nonlinear cou-plings.When the potential function in couplings is periodic,we obtain infinitely many solutions for the system by applying index theory,When the potential function is non-periodic,we apply Nehari manifold method and concentration compactness principle to prove the asymptotic behaviour of ground state solutions.In Chapter 6,we apply Nehari manifold method and decomposition of Palais-Smale sequence to prove the existence and asymptotic behaviour of ground state solutions for a fractional Schr?dinger-Poisson system with Hardy potentials in R~3.