Existence And Multiplicity To Solutions For Nonlinear Schr(?)dinger-poisson Systems

Nonlinear partial differential equations are the subject with important research value and a broad application background in physics,chemistry;biology and other fields of science.They usually arise in the natural science and engineering areas for the purpose of application.A large number of researchers have devoted much attention to these equations for a long time.Schrodinger equation is one of the most general equations,the existence,nonexistence and multiplicity of solutions have also attracted extensive attention in recent years.In this paper,we use variational methods and critical point theorem to discuss the properties of solutions for two kinds of more general Schrodinger-Poisson systems.The thesis consists of three sections.Chapter 1 is the preface.In Chapter 2,we study the existence of solutions for the following Schrodinger-Possion system,where ??(1,3),?,?>0 are parameters.For h,k,l,we assume the following hypotheses hold:(Hh)h?L3/2(R3)\{0},and h(x)?0 for all x?R3;(Hk)k?C(R3)changes sign on R3 and lim|x|??k(x)=k?<0;(Hl)l?L6/(5-?)(R3)\{0} and l(x)?0 for all x?R3.Under the hypothese(Hh),there exists an eigenvalues sequence {?n} of-?u+u=?h(x)u,x?R3,with 0<?1<?2?…??n?…,where and each eigenvalue is of finite multiplicity.The associated normalized eigenfunctions are denoted by e1,e2,...,en,....To go further,we definewhere ||·|| denotes the norm of the space H1(R3).By seeking the local minimizer of the energy functional on the Nehari manifold,we have the following conclusions.Theorem 2.1.1 Assume the hypotheses(Hh),(Hk)and(Hl)hold.Then for every??(0,?1)and .the system above has at least one solution in H1(R3)x D1,2(R3).Theorem 2.1.2 Assume the hypotheses(Hh),(Hk)and(Hl)hold.Then for any given?'?(0,?*),there exists ?>0 such that the system above has at least one solution whenever??(?1,?1+?)and ??(0,?'].In Chapter 3.we study the multiplicity to solutions for the following Schrodinger-Possion system where ? ?(2,3),For V,l,f,we assume the following hypotheses hold:(V)V?C(R3,R),infx?R3V(x)=V0>0,and there exists v0>0 such that lim|y|??meas{x?R3:|x-y|?v0,V(x)?M}=0,M>0;(l)l?L6/(5-?)(R3),and l(x)?0 for all x?R3;(f1)f?C(R3×R,R)and exists C>0 such that |f(x,t)|?C(|t|+|t|p-1)for some p?(4,6);(f2)lim|t|??F(x,t)/|t|2?=? for all x?R3,where F(x,t)=?t0f(x,s)ds;(f3)Let F=1/2?f(x,t)t-F(x,t),there exist C'>0,r0>0 such that if |t|?r0,then F?-C'|t|2 uniformly for x?R3;(f4)f(x,-t)=-f(x,t),(x,t)?R3× R.Using the Fountain theorem,we have the following conclusion.??3.1.1 Assume the hypotheses(V),(l),(f1)-(f4)hold.Then the system above possesses infinitely many nontrivial solutions .