LAUSR

This paper is concerned with the existence of positive solutions for the fourth order Kirchhoff type problem { ∆u− (a+ b ∫ Ω |∇u|dx)△u = λf(u(x)), in Ω, u = △u = 0, on ∂Ω, where Ω ⊂ R (N ≥ 1) is a bounded domain with smooth boundary ∂Ω , a > 0, b ≥ 0 are constants, λ ∈ R is a parameter. For the case f(u) ≡ u , we use an argument based on the linear eigenvalue problems of fourth order elliptic equations to show that there exists a unique positive solution for all λ > Λ1,a , here Λ1,a is the first eigenvalue of the above problem with b = 0 ; For the case f is sublinear, we prove that there exists a positive solution for all λ > 0 and no positive solution for λ < 0 by using bifurcation method.