Existence Of Solutions For Kirchhoff Type Equation With Critical Growth In High Dimension

This thesis studies the following Kirchhoff type equation-(a+b?_R^N|(?)u|~2dx)(?)u+V(X)u=?f(x,u)+|u|^2*-2u,u(?)H~1(R^N),where dimensionN?4,a,b,?>0,V is a positive potential.WhenN=4,f(x,u)=g(x)|u|^q-2u,for q=2,we obtain that the above equation has a positive ground state by employing the mountain pass theorem,energy estimation,Nehari manifold and a deduction of Ekeland variational principle;for(?)(2,4),applying Jeanjean monotonicity trick,Pohozaev identity,global compactness lemma,we prove the the existence of ground state for the above equation.Secondly,when N?5,considering the general nonlinear term f(x,u),we prove the existence of two nontrival solutions for the above equation by defining fibering maps and extremal parameters,combining with the weakly lower semi-continuity of functional and(PS)condition.Finally,utilizing a new method of modifying the minimizing sequence,we solve a problem coming from Remark 1.2 in Xu[J.Math.Anal.Appl,2020].