Existence results of positive solutions for Kirchhoff type biharmonic equation via bifurcation methods

Abstract

This paper is concerned with the existence of positive solutions for the fourth order Kirchhoff type problem { ∆2u − (a + b ∫ Ω |∇u| 2 dx)△u = λf(u(x)), in Ω, u = △u = 0, on ∂Ω, where Ω ⊂ R N (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.