On The Existence And Properties Of Solutions For Some Nonlocal Elliptic Equations

In this paper,we mainly study the existence and properties of solutions for three kinds of nonlocal elliptic equations and systems,including the Choquard equation,nonlinear Choquard equation and Schr(?)dinger-Poisson system driven by fractional Laplacian.The thesis consists of five chapters:In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some preliminary results and notations used in the whole thesis.In Chapter Two,we prove that each positive solution of-?u=(Ia*|u|2a*)|U|2a*-2u,u ? D1,2(RN)(Q1)is radially symmetric,monotone decreasing about some point and has the form ca(t/t2+|x-x0|2)N-2/2,where N ? 3,? ?(0,N),2a*:=N+a/N-2 is the upper Hardy-Littlewood-Sobolev critical exponent,t>0 is a constant and ca>0 depends only on a.We also study the following nonlinear Choquard equation-?u+ V(x)u =(Ia*|u|2a*)|u|2a*-2u,u ? D1,2(RN).(Q2)By using Lions' Concentration-Compactness principle,we obtain a global compact-ness result,i.e.we give a complete description for the Palais-Smale sequences of the corresponding energy functional.Adopting this description,we succeed in proving the existence of at least one positive solution if ||V(x)|| LN/2 is suitable small.This re-sult generalizes the result for semilinear Schr(?)dinger equation by Benci and Cerami(J.Funct.Anal.,88,90-117(1990))to Choquard equation.In Chapter Three,we consider the following nonlinear Choquard equation driv-en by fractional Laplacian(-?)su + ?V(x)u =(Ia*F(u))f(u),x ? RN,(Q3)where s ?(0,1),N?2,? ?((N-4 s)+,N)and d is a positive parameter,V(x)is a nonnegative continuous potential function.By variational methods,we prove the existence of ground state solution which localizes near the bottom of potential well int(V-1(0))as ? large enough.The results mentioned above have been published on Math.Methods Appl.Sci.,41,1145-1161(2018).In Chapter Four,we consider the existence of sign-changing solutions for the following fractional Schr(?)dinger-Poisson system{(-?)su + V(x)u + ??(x)u = f(u),x ? R3,(-?)t?=u2,x ? R3,(Q4)where s,t ?(0,1),? is a positive parameter,V(x):R3?R+ is a continuous potential function.Since multiple nonlocal terms are involving in the system,some new techniques are applied to prove the existence of a least energy sign-changing solution ??.Moreover,we prove that the energy of any sign-changing solution to this system is strictly larger than twice of the ground state energy.Furthermore,the asymptotic behavior of as ?? 0+ is also analyzed.These results generalize the results for Schr(?)dinger-Poisson system by Wang and Zhou(Calc.Var.Partial Differential Equations.,52,927-943(2015)),Shuai and Wang(Z.Angew.Math.Phys.,66,3267-3282(2015))to fractional Schr(?)dinger-Poisson system and have been accepted by Appl.Anal.In Chapter Five,we consider the existence of positive solutions for the following fractional Schr(?)dinger-Poisson system{?2s(-?)su+V(x)u+?(x)u =k(x)f(u)+|u|2s*-2u,x ? R3,?2s(-?)s?=u2,x ? R3,(Q5)where s ?(3/4,1),? is a small and positive parameter,V(x),K(x)are nonnegative potential functions.2s*is the critical exponent with respect to fractional Sobolev embedding theorem.Under some suitable conditions on the nonlinearity f and po-tential functions V(x),K(x),we prove that for ? small,the system(Q5)has a pos-itive ground state solution concentrating around a concrete set related to V(x)and K(x).This result generalizes the result for fractional Schr(?)dinger-Poisson system with subcritical exponent by Yu et al.(Calc.Var.Partial Differential Equations,56,(2017))to critical exponent.Moreover,when V(x)attains its minimum and K(x)attains its maximum,we also obtain multiple solutions by Ljusternik-Schnirelmann theory.