Existence Of Solutions For Some Kinds Of Kirchhoff Type Equations

In this paper,we study the generalization of Kirchhoff type equation by using the variational method where ? is bounded domian in RN,a,b>0,?(x,u)?C1(?×R,R).Firstly,we consider the existence of solutions for p-kirchhoff type equations by Morse theory Supposed that the existence of the equatio-(a+b ??|?u|pdx)?pu=f(x,u),then the equation(0-2)has multiple solutions by Morse theory.Using Morse theory needs to satisfy PS condition,but since we weaken the condition f(x,u),such that the embedded compactness does not exist.In order to prove PS condition,we use Vitali convergence theorem to overcome this problem.Secondly,we study the existence of solutions for Kirchhoff type equations with pertur-bation term where ? is bounded domian in R3,a,b>0,f(x,u)?C1(?×R,R),h(x)? L2(?),In this section,we consider that f(x,u)has a more general growth condition.By constructing Nehari manifolds with and without perturbations.When the perturbations are small enough,we obtain that there are two different nontrivial solutions to(0-3)by using the critical point theory.Finaly,we consider the existence of solutions for a class of Kirchhoff type equations with nonlocal and critical terms where ? is a smooth bounded domian in RN.a,b>0,??R,N>5,q?[1,2*-1),2*=2N/N-2.Supposed that the potential function V(x)satisfies V ?(?,R),(?).By constructing the limit function of the Fibering map of the PS sequence and the concentration minimizing sequence,there are two different nontrivial solutions to(0-4).