The Existence Of Solutions For Several Classes Of Nonnocal Elliptic Equations(systems)

Variational method is one of the important fundamental methods in the nonlin?ear functional analysis.The basic idea is that the problem of differential equations convert to the critical point of corresponding functional problem.This dissertation will be concerned with the existence of ground state solutions for gauged nonlinear Schrodinger equations,the existence of solutions for fractional Schrodinger-Poisson systems involving a Bessel operator,the multiplicity and concentration results for fractional Schrodinger-Poisson systems involving a Bessel operator,the existence of least energy solutions for fractional Schrodinger-Poisson systems.These nonlo-cal problems have wide applications of physical backgrounds,such as mathematical physics and quantum mechanics.There are five chapters in the dissertation.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 are concerned with the existence of ground state solutions and minimal energy solutions for the following nonlinear Schrodinger equation with critical exponential growth and subcritical exponential growth of gauged type where ?>0,?>0 is a constant representing the strength of interaction potential,By using variational methods,we prove the existence of ground state solutions and least energy solutions for(E1)with critical exponential growth and subcritical ex-ponential growth,respectively.The main results in this chapter extend the main results in Byeon-Huh-Seok(J.Funct.Analysis.2012)and Ji-Fang(J.Math.Anal.Appl.2017).In Chapter Three,we study the existence of nontrivial solutions for the following fractional Schrodinger-Poisson systems where(I-?)s is the Bessel operator and(-?)t is a fractional Laplacian operator for s ?(3/4,1)and t ?(0,1),respectively.By using the variational methods and perturbation argument,we prove the existence of nontrivial solutions for(E2),which extends the main results in Liu-Wang(J.Differential Equations,2014).The main results in this chapter have been published in(Comput.Math.Appl.2018).In Chapter Four,we study the multiplicity and concentration of nontrivial solutions for the following concave-convex elliptic systems:where 1<q<2,the parameter ?>0,(I-?)s is the Bessel operator and(-?)t is a fractional Laplacian operator for s ?(3/4,1)and t ?(0,1),and V ? C(R3,R)is a steep potential well.We use the Mountain-Pass theorem and Ekeland variational's principle to prove the multiplicity and concentration of nontrivial solutions(E2),which extends the mains results in Secchi(Compl.Var.Elliptic Equations,2016).The main results in this chapter have been published in(Math.Method Appl.Sci.2018).In Chapter Five,we consider the existence of nontrivial least energy solutions for the following nonlinear fractional Schrodinger-Poisson systems involving:where(-?)? is the fractional Laplacian for ?=s,t?(0,1)with s<t and 2s+2t>3.When the potential V? C(R3,R)satisfies some suitable conditions,we use the variational methods and monotone trick to prove the existence of nontrivial least energy solutions for(E4),which extends the main results in Teng(J.Differential Equations,2016).The main results in this chapter have been published in(J.Math.Phys.2018).