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=λf（x,u）+|u|^2*-2u,u（?）H~1（R^N）,where dimensionN≥4,a,b,λ>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].