The Existence And Multiplicity Of Constrained State Solutions For A Class Of Schr(?)dinger-poisson Systems

Nonlinear partial differential equations have a wide range of physical backgrounds.It has important value in theory and application and therefore it is studied by scientific researchers.Schr（?）dinger-Poisson equation is a kind of very important nonlinear partial differential equa-tion,and there have been many research results in recent years.In this thesis,we study the existence and multiplicity of constrained state solutions for a class of Schr（?）dinger-Poisson systems with a general nonlinear term.Firstly,we prove the existence of constrained state solutions for a class of Schr（?）dinger-Poisson systems with a general nonlinear term in R^N by minimizing sequence method,vanishing lemma,Br′ezis-Lieb lemma and some analytical skills.Next,we prove the multiplicity of constrained state solutions for a class of Schr（?）dinger-Poisson systems with a general nonlinear term in Rby minmax method and Ekeland variational prin-ciple.The main theoretical bases are minimizing sequence method,vanishing lemma,varia-tion principle,minmax method,Hardy-Littlewood-Sobolev inequation,Gagliardo-Nirenberg inequation,Pohozaev identity,Nehari manifold and some analytical methods.This thesis is divided into 4 chapters.In chapter 1,we introduce the development of variational methods and the recent research achievements of Schr（?）dinger-Poisson systems.Moreover,we state the research work and main results of this thesis and give the preliminary knowledges used in this thesis.In chapter 2,we study the existence of constrained state solutions for the following Schr（?）dinger-Poisson systemswith the conditionS_c={u∈H¹（R³）:‖u‖₂²=c,c>0},whereλ∈R is lagrange multiplier and f,K satisfy the following conditions:（F1）f∈C（R,R）,and there exist C₀>0 and q∈（3,10/3）such that|f（t）|≤C₀(1+|t|^q-1),t∈R;（F2）there exists μ∈（3,10/3）such that f（t）t≥μF（t）>0,t∈R;（F3）there exists p∈（2,8/3）such that;（K1）K∈（R³,R⁺）and ,x∈R³;（K2）for every y∈R³,t^2（μ-3）K（y）≥K（ty）,t≥1,where f（t）:=∫?₀^tf（s）.Firstly,we transform the existence problem of constrained state solutions of the above Schr（?）dinger-Poisson systems into the critical point problem of energy functional on the con-straint.Secondly,we prove that the infimum of the energy functional on the constraint exists and is less than zero by Gagliardo-Nirenberg inequality and then we find a miniminzing se-quence.Finally,we obtain the normalized minimizer of energy functional on the constraint by vanishing lemma,Br′ezis-Lieb lemma,embedding theorem and interpolation inequality and then obtain the existence of constrained state solutions for the systems.In chapter 3,we consider the multiplicity of constrained state solutions for the following Schr（?）dinger-Poisson systemswith the conditionS_r（c）:={u∈H_r¹（R^N）:‖u‖₂²=c,c>0},where N=3,4,λ∈R is lagrange multiplier and f satisfies the following conditions:（F4）f∈（R,R）is a odd function,f（t）t≥0 and there exist>0 and q∈（2+4/N,2^*）such that|f（t）|≤C(1+|t|^q-1),t∈R;（F5）and;（F6）（f（t）t-2F（t））/(|t|^p-1)t)is nondecreasing on（0,+∞）,where p=2+4/N and F（t）:=∫?₀^tf（s）.Firstly,we transform the existence problem of constrained state solutions of the above Schr（?）dinger-Poisson systems into the critical point problem of energy functional on the con-straint.Secondly,we obtain the minmax sequence by minmax method.Finally,we find the corresponding PS sequence by Ekeland variational principle and prove its compactness.Then we obtain a list of constrained state solutions whose energy level is strictly less than zero.In chapter 4,we summarize the conclutions of this thesis and give the futher research directions.