Sign-changing and multiple solutions of Kirchhoff type problems without the PS 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 integral(Omega)\del u\(2))Delta u = f(x, u) in Omega, on partial derivative Omega. u = 0 (C) 2008 Elsevier Ltd. All rights reserved.