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 = 0 on ∂ Ω .