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

In this thesis,we mainly study the existence,multiplicity and properties of nontrivial solutions for some nonlocal elliptic equations and systems.There are five chapters in this thesis.In Chapter One,we summarize the background and the current research status of the related nonlocal 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 consider the following singularly perturbed nonlocal elliptic problemwhere e>0 is a parameter,a>0,b>0 are constants,??(0,3),p ?(2,6-?a),W?(x)isa convolution kernel and F(x)is an external potentials satisfying some conditions.By us-ing variational methods,for ? sufficiently small,we establish the existence,concentration and decay property of positive ground state solutions for the above equation,In Chapter Three,we study the following Kirchhoff-type equation with general Hartree-type nonlinearitywhere ?>0 is a small parameter,a,b>0 are constants,??(0,3)and V(x)?(R3,R)is an external potential,F ? C1(R,R)and f = F'.Under some suitable assumptions on V(x)and f,by using the penalization method,we prove that for ?>0 small enough there exists a family of positive solutions which concentrate at the minimum point of the potential V(x)as ??0.In Chapter Four,we study the following coupled nonlinear fractional Laplacidan system where(-?)?,??(0,1)denotes the usual fractional Laplacian operator,N>2?,?>0 is a coupling parameter.Under very weak assumptions on the nonlinear terms f and g,we establish the existence of positive ground state vector solutions and positive higher energy vector solutions respectively for the fractional Laplacian systems by using vari-ational methods.In addition,we also study the asymptotic behavior of these solutions as the coupling parameter ?? 0.Finally,in Chapter Five,the following Kirchhoff-Schrodinger-Poisson system is stud-ied:where a>0,b>0 are constants and ?>0 is a parameter,f?C(R,R).Without assuming the Ambrosetti-Rabinowitz type condition and monotonicity condition on f,we establish the existence of positive solutions for the above system by using variational methods combining a monotonicity approach with a delicate cut-off technique.We also study the asymptotic behavior of solutions with respect to the parameter ?.In addition,we obtain the existence of multiple solutions for the nonhomogeneous case corresponding to the above problem.