Existence Of Least Energy Sign-changing Solutions For Kirchhoff-type Problems With Critical Growth

In this paper,by taking advantage of the variational methods and some sig-nificant analysis techniques,we consider the existence of least energy sign-changing solutions for two classes of Kirchhoff-type problems with critical growth.Firstly,we study the existence of least energy sign-changing solutions for the following autonomous Kirchhoff-type problem with critical growth:-(a+b∫_R³|▽u|²dx)Δu+u=f（u）+|u|⁴u,x（?）R³,（SK₁） where a>0 and b>0 is small enough.Under some suitable assumptions on the subcritical nonlinearity f（?） C（R,R）,we obtain that the problem（SK₁）has a least energy sign-changing solution u_bwith exactly two nodal domains,and its energy is strictly larger than twice that of ground state solution.Secondly,we study the existence of least energy sign-changing solution for the following critical Kirchhoff-type problem with vanishing potential:-(a+b∫_R³|▽u|²dx)Δu+V（x）u=|u|⁴u+λK（x）f（u）,x（?）R³,（SK₂）where a>0,b>0,the subcritical nonlinearity f（?） C（R,R）,and K（x）,V（x）are positive and continuous functions vanishing at infinity.Combining the variational methods with λ parameter is large enough,we prove that the problem（SK₂）has a least energy sign-changing solution u_λ with exactly two nodal domains.