Ground State Solution And Multiple Solutions To Two Kinds Of Schr(?)dinger-kirchhoff Equations

The research object of this paper is Schr(?)dinger-Kirchhoff equations.By means of variational method we obtain the multiplicity of the solution and the existence of the ground state solution respectively.Firstly,we mainly introduce the background and current situation of the research of Schr(?)dinger-Kirchhoff equation,the preparatory knowledge,the main notation and the main results of this paper.Secondly,we investigate multiple solutions for a class of Schr(?)dinger-Kirchhoff equations with indefinite potential and concave and convex nonlinearities.we first obtain the energy functional of the corresponding equation through the variational method,and then prove that the corresponding functional is coercive and satisfiess the conditions of(PS).Finally,we obtain multiplicity of solutions by using Clark’s theorem and Dual fountain theorem.Finally,by using a version of Trudinger-Moser inequality and the Nehari manifold technique,we study the existence of ground state solutions for a class of Schr(?)dinger-Kirchhoff equations with vanishing potential and critical exponential growth.First,we get the corresponding energy functional of the equation,then we get two key inequalities by proving some necessary lemmas,and finally we get the existence of the ground state solution according to the two inequalities and the definition of the ground state solution.