Existence Of Solutions For Several Kinds Of Schr?dinger-Poisson Systems

With the development of nonlinear science,nonlinear partial differential equations close-ly link mathematical theory with practical application.In addition,a large number of non-linear partial differential have emerged in physics,chemistry,mechanics and other fields,such as Schr?dinger-Poisson system.The system appears in quantum mechanics models and in semiconductor theory,which is a model to describe solitary waves for the nonlinear Schr?dinger equations interacting with the electrostatic field.Moreover,the existence and multiplicity of solutions about Schr?dinger-Poisson system have always been the focus of scholars all the time.In this paper,we use variational methods,such as Hardy inequali-ty,logarithmic Sobolev inequality,the mountain pass theorem,concentration-compactness principle and symmetry mountain pass theorem,to discuss the solutions of the Schr?dinger-Poisson problems under different assumptions.The thesis consists of three sections.In Chapter 1,we consider the following Schr?dinger-Poisson system with Hardy term and logarithmic nonlinearity where ? is a bounded smooth domain in R3,0??<?:=1/4 and ? denotes the best Hardy constant.The main purpose of this paper is to study the existence of nontrivial solutions for the above system when the nonlinear term f satisfies appropriate assumptions.Specifically,the nonlinear term f satisfies(f1)function f?C(R,R),and f(0)=0;(f2)limt?0f(t)/t=0;(f3)There exists C>0 and p ?(2,6)such that for every t? R,(f4)limt??F(t)/t4=?,where F(t)=?0t f(s)ds,t?R;(f5)there exist ?(0,4)and C1,C2>0 such that for every t?R,f(t)t-4F(t)?C1|t|2+?-C2t2.We will use Hardy inequality,logarithmic Sobolev inequality and the mountain pass theorem to prove that the above system possesses a nontrivial solution.In Chapter 2,by using concentration-compactness principle and symmetry mountain pass theorem,we study the multiplicity of solutions for the following Schr?dinger-Poisson system with critical exponent where ?(?)R3 is a bounded smooth domain,q ?[4,6)and ?,?>0 are two positive parameters.In Chapter 3,we consider the following Schr?dinger-Poisson system with critical and supercritical nonlinear terms where p ?(4,6),q?(6,+?)and b ? C(R3,R+)is a radial potential function,which satisfies the following conditions:(Bi)there exists a r>0 such that lim|x|?0b(x)/|x|?=b0?0;(B1)b?=lim|x}??b(x)<?.According to the property of potential function at zero point,we get a compact em-bedding result,and then combining with the estimate on the mountain pass energy level,the mountain pass theorem and Nehari manifold methods,we obtain the existence of ground state solutions for the above system.