Existence of solutions for fourth order elliptic equations of Kirchhoff type

This paper is concerned with the existence of nontrivial solutions for a class of fourth order elliptic equations of Kirchhoff type (1) { Δ 2 u − λ ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u = f ( x , u ) , in Ω , u = 0 , Δ u = 0 , on ∂ Ω , where a > 0 , b ≥ 0 are constants, and λ > 0 is a parameter. We will show that there exists a λ ∗ such that (1) has nontrivial solutions for 0 < λ < λ ∗ by using the mountain pass techniques and the truncation method.