Existence And Uniqueness To Solutions For A Kind Of Nonlinear Kirchhoff-schr(?)dinger-Poisson Systems

Nonlinear partial differential equations are important branch of modern mathematics.They are the general problems in the natural science and engineering ares.No matter in theoretical or practical application,they both have great significance and value,and a large number of researchers have devoted much attention to these equations for a long time.Moreover,for Kirchhoff equations and Schr(?)dinger equations as the most fundamental equations in the partial differential equations,the existence,uniqueness and multiplicity of solutions have also attracted extensive attention in recent years.In this paper,we use variational methods,such as the Mountain pass theorem and Global compactness lemma,to discuss the existence and uniqueness of solutions for two kinds of Kirchhoff-Schr(?)dinger-Poisson systems.The thesis consists of three sections.Chapter 1 is the preface.In Chapter 2,we consider the following Kirchhoff-Schr(?)dinger-Possion system where ? C R3 is a smooth bounded domain,a,b ? 0 with a + b>0,?,??R+:=[0,?).For f,h and g,we assume the following hypotheses.hold:(f0)f ?((0,?),R+)satisfies that there exists ?>0 such that f is nonincreasing on(0,?],?0? f(s)ds<?,and there exist ?,? ?(0,1)such that(f1)there exists a constant k ?(0,aS|h|3/2-1)such that where S is the best Sobolev constant.(f2)f ?((0,?),R+)is nonincreasing on(0,?)and ?01 f(s)ds<?.Moreover,there exists ? ?(0,1)such that(h0)h ? L6/(5-?)(?)satisfies h(x)>0,a.e.x??;(hi)h E L3/2(?)satisfies h(x)>0,a.e.x??;(g)g ? C(R+,R+)and there exists c>0 such that g(s)? c(s + s5),s ? R+.Using the Variational method,we have the following theorems.Theorem 2.1.1.If a,6 ? 0 with a + b>0 and the assumptions(h0),(f0)and(g)hold,then the system above possesses a solution for any ?,??R+.Moreover,this solution is a global minimizer of the energy function for the system above.Theorem 2.1.3.Let a>0 and the assumptions(h1),(f0),(f1)and(g)hold.Moreover,assume that g is nondecreasing on R+,then the system above possesses a unique solution for any ?,? E R+.Theorem 2.1.5.If the assumptions(h0),(f2)and(g)hold and g is nondecreasing on R+,then the system above possesses a unique solution for any ?,?? R+.In Chapter 3,we consider the following Kirchhoff-Schr(?)dinger-Possion system where a>0,b ? 0 are constants,?? 0 is small enough.f,V satisfy the following conditions:(f3)f? C1(R3,R+),f(s)/s3 is increasing on(0,?)and lims?? f(s)/s3=?;(f4)lims??f'(s)/s4 = 0;(V)V ? C(R3,R+),lim|x|?? V(x)=V? exists and V? is positive.Moreover,V satisfies V(x)? V?,x ? R3.Using the Mountain pass theorem and Global compactness lemma,we obtain the fol-lowing results.Theorem 3.1.1.If conditions(f3)-(f4)are satisfied and V = V?.then the system above has a positive ground state solution.Theorem 3.1.2.If conditions(f3)-(f4)and(V)are satisfied,then the system above has a positive ground state solution.