Existence Of Constrained State Solution For A Class Of Nonlinear Kirchhoff Equations

Nonlinear partial differential equation usually arises in the natural science and engi-neering field,and has a wide range of background.No matter in theory or practice,it is of great significance and value.Because it can well explain the important phenomenon of nature and solve the nonlinear problems effectively,a large number of science researchers have paid attention to the problems for a long time.Kirchhoff equation is the most basic and important equation of partial differential equations.Since the appearance of nonlocal item |?u|22?u,these equations have been widely studied by scholars in recent years.In this paper,by estimating the corresponding energy functional of Kirchhoff equations with regular L2(R4)norm and using the Gagliardo-Nirenberg inequality and the proof of subadditivity conditions in four dimensional space,we discuss the minimization problem of Kirchhoff e-quations which are restricted in the L2(R4)norm and obtain the existence of minimizers of the energy functional.In the second chapter,we consider the following Kirchhoff equation-(a+b(?)R4|?u|2)?u+V(x)u=c|u|p-2u+?u,By establishing some delicate estimates on the least energy of the GP functional,we study following minimization problem eVc:=inf u?SV EVc(u)where a,b>0 are constants and EVc(u)=a/2|?u|22 + 1/2(?)R4V(x)u2+b/4|?u|24-c/p|u|pp,SV:= {u?HV:|u|2 = 1}.And we obtain the existence of constrained minimizers of nonlinear Kirchhoff equations in two cases V = 0 and V? 0.In the third chapter of the paper,we consider the following Kirchhoff equation-(a+b(?)R4 |?u|2)?u = |u|p-2u+?u,x?R4,(1)where ??R.And we discuss existence of constrained minimizers with a prescribed L2(R4)-normalized for Kirchhoff equations,with the Gagliardo-Nirenberg inequality and the proof of the strict subadditivity condition,namely,following minimization problem mc2:= inf u?Sc E(u),(2)where E(u)=a/2(?)R4|?u|2+b/4((?)R4|?u|2)2-1/p(?)R4|u|p,Sc = {u ?H1(R4):|u|2=c>0}.The problem consideration is associated to the research of standing waves,namely,solutions having the form solutions having the form for ??R,u(x)is solution of equation(1)of the nonlinear Kirchhoff equation-i(?)t?=(a+b(?)R4|??|2)??+|?|p-2?.(3)And we obtain existence and orbital stability of constrained minimal standing waves for nonlinear Kirchhoff equations.