| In this thesis,we mainly study two classes of nonlocal elliptic equations in whole space,involving quantum Zakharov system and Choquard equation.Our aim is to investigate the ex-istence and multiplicity of solutions for this equations when the attractive or competing effect of the nonlocal term with the nonlinear perturbation happens.This thesis consists of four chapters:In Chapter 1,we first introduce the research background and meaning about two classes of nonlocal elliptic equations.And then we give research status and development of the related problems and state some definitions and preliminary lemma.Finally,we show the mainly re-search contents and methods in this thesis.In Chapter 2,we study a class of stationary quantum Zakharov system with an attractive perturbation as followswherelλ>0,μ∈R,p>1,f(x),K(x) and V(x) are nonnegative functions.By using the Nehari manifold,we prove the existence and multiplicity of nontrivial solution,depending on the parametersl,mand p.Specially,for higher competing perturbation(μ<0,p>4),such problem can not be studied via the common arguments in variational methods,since Pal-ais-Smale sequences may not be bounded.As a result,we try to overcome the difficulty by us-ing a new constraint approach.In Chapter 3,we study a class of Choquard type equations with a competing perturbation as followswhere N≥3,λ>0, a parameter,I_a is the Riesz potential,K(x) and V(x) are nonnegative functions,2<q<2~*and (N+α)/N<p<(N+α)/(N-2).We are inter-ested in the existence and multiplicity of positive solutions under the different relationship be-tween p and q.In particular,when 2p<q,variational methods can not be applied in a standard way,even restricting the energy functional on the Nehari manifold,because Pal-ais-Smale sequences may not be bounded.A new constraint approach is employed to prove the existence and multiplicity of positive solutions for above equations.In Chapter 4,the main results are summarized and we present some relevant problems. |
