On Fourth-Order Elliptic Equations of Kirchhoff Type with Dependence on the Gradient and the Laplacian

We consider a nonlocal fourth-order elliptic equation of Kirchhoff type with dependence on the gradient and Laplacian Delta(2)u - (a + b integral(Omega) |del u|(2)dx)Delta u = f(x,u,del u,Delta u), in Omega, u = 0, Delta u = 0, on partial derivative Omega, where a,b are positive constants. We will show that there exists b* > 0 such that the problem has a nontrivial solution for 0 < b < b* through an iterative method based on the mountain pass lemma and truncation method developed by De Figueiredo et al., 2004.