Research On The Existence And Multiplicity Of Solutions For The Klein-Gordon-Maxwell System

In our thesis,we study the existence of infinity many solutions and ground state solutions,the multiplicity of solutions and the convergence property of solu-tions for the Klein-Gordon-Maxwell system.The method consists of variational methods and some analysis techniques.The Klein-Gordon-Maxwell system has strong physical background.The general form is as follows (?) where w>0 denotes the phase,u,?:R3?R are unknown,f is the nonlinearity.When V(x)is constant,we call it autonomous.When V(x)is not constant,we call it non-autonomous.Firstly,we consider the following non-autonomous Klein-Gordon-Maxwell system with steep potential well (?)where a(x)satisfies the following conditions(a1)V?C(R3,R),a>0 and ?:=a-1(0)has nonempty interior and smooth boundary.(a2)There is M>0 such that?({x ?R3:a(x?M})<?,(?)where ? stands for the Lebesgue measure.When ???,??(x)+1 is called the steep potential well.We study the existence of ground state solutions,the multiplicity of solutions and the convergence property of solutions for the the system(0.3)in the Chapter 2,3,4.In Chapter 2,we research the the existence of ground state solutions and the convergence property for the system(0.3)with the nonlinearity f being super-linear and subcritical.Based on the mountain pass lemma and the definition of ground state solution,we get the existence of ground state solution.In Chapter 3,we investigate the the existence of the multiplicity of solutions for the system(0.3)with the nonlinearity f being subcritical.From the idea of the fountain theorem,a sequence of solutions can be gained.Without global and local compactness,we can tell the difference of multiple solutions from their norms in Lp(R3).In Chap-ter 4,we study the the existence of ground state solutions and the convergence property for the system(0.3)with the nonlinearity f being asymptotically linear and subcritical.By using the saddle point theorem and the definition of ground state solution,we obtain the existence of ground state solution.Secondly,we study the following autonomous Klein-Gordon-Maxwell system#12 where m>0 is mass.In Chapter 5,we study the infinitely many high energy radial solutions for the system(0.4)with the nonlinearity f being superlinear and subcritical.By using some technique related to Pohozaev identity,we construct bounded Palais-Smale sequences to obtain the existence of infinitely many solu-tions.