Existence And Properties Of Solutions For Kirchhoff Elliptic Equations And Kirchhoff Systems

In this paper,we mainly study the existence and asymptotical behavior of solu-tions to Kirchhoff equation and Kirchhoff system in R3.The main work is divided into the following four parts.First,we consider the existence of multiple solutions to Kirchhoff equation-(a+b?R3|?u|2dx)?u+u=|u|p-2u,u?H1(R3),where a>0,b? 0 and 2<p<6.By constructing a new variational identity of Pohozaev type and a constraint set,we are able to prove the existence of radially symmetric solution and the non-radially symmetric solution to the problem.Moreover the non-radially symmetric solution u(x1,x2,x3)is radially symmetric with respect to(x1,x2)and odd with respect to x3.Secondly,for a stationary version to a class of wave system u,v?H1(R6),by establishing a variant variational identity and constraint set,we prove that for as>0,bs? 0(s=1,2),d? 0,b1+b2+d?0 and p>1,q>1 with 2<Q:=p+q<6,the system admits a ground state solution,a positive radially symmetric solution and a non-radially symmetric solution respectively.The non-radially symmetric solution u(x1,x2,x3)and v(x1,x2,x3)are radially symmetric with respect to(x1,x2)and odd with respect to x3.Moreover,for any fixed a1>0 and a2>0,as b62+b22+c2?0,this radially symmetric solution converges to a radially symmetric solution to-a1?u+u=p/Q|u|p-2u|v|q,-a2?v+v=q/Q|u|p|v|q-2v,u,v?Hr1(R3).Thirdly,for the linearly coupled Kirchhoff-type system u,v?H1(R3),we prove that for as>0,bs?0(s=1,2),d?O,b1+b2+d?0,2<p<6 and 0<?<1,the system admits a ground state solution,a positive radially symmetric solution and a non-radially symmetric solution respectively.The non-radially symmetric solution u(x1,x2,x3)and v(x1,x2,x3)are radially symmetric with respect to(x1,x2)and odd with respect to x3.Moreover,for any fixed a1>0 and a2>0,as b12+b22+c2?0,this radially symmetric solution converges to a radially symmetric solution to-a1?u+u=|u|p-2u+?v,-a2?v+v=|v|p-2v+?u,u,v?Hr1(R3).Finally,by establishing a new variational constraint,we prove that for a1>0,a2>0;b1>0,b2>0,d?0,?>1,?>1 with 2<?+?=p<6;and for suitable?,the following system admits a ground state solution(u,v)?H1(R3)×H1(R3)with u>0 and v>0.This result generalizes partial works of Lin and Wei(Commun.Math.Phys.255(2005),629-653)and Sirakov(Commun.Math.Phys.271(2007),199-221).In the case of a1=a2 and b1=b2,several existence and nonexistence of solutions with special forms are also investigated.