Existence of solutions for fourth order elliptic equations of Kirchhoff type

•The paper deals with the existence of solutions of a fourth order stationary Kirchhoff equation depending on one parameter λλ.•It is possible to weaken the assumption on bb in the Kirchhoff function M(t)=a+btM(t)=a+bt, and require b≥0b≥0.•The techniques used are quite unusual.

This paper is concerned with the existence of nontrivial solutions for a class of fourth order elliptic equations of Kirchhoff type equation(1){Δ2u−λ(a+b∫Ω|∇u|2dx)Δu=f(x,u),inΩ,u=0,Δu=0,on∂Ω, where a>0a>0, b≥0b≥0 are constants, and λ>0λ>0 is a parameter. We will show that there exists a λ∗λ∗ such that (1) has nontrivial solutions for 0<λ<λ∗0<λ<λ∗ by using the mountain pass techniques and the truncation method.