Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition

The existence of nontrivial solutions of Kirchhoff type equations is an important nonlocal quasilinear problem; in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4-superlinear Kirchhoff type problems without the P.S. condition, one concerning the existence of a nontrivial solution and the other one concerning the existence of sign-changing solutions and multiple solutions, {−(a+b∫Ω|∇u|2)△u=f(x,u)in Ω,u=0on ∂Ω.