Global Bifurcation Result for the p-Biharmonic Operator

Abstract

We prove that the nonlinear eigenvalue problem for the p-biharmonic operator with p > 1, and Ω a bounded domain in ℝN with smooth boundary, has principal positive eigenvalue λ1 which is simple and isolated. The corresponding eigenfunction is positive in Ω and satisfies ∂u/∂n < 0 on ∂Ω, ∆u1 < 0 in Ω. We also prove that (λ1, 0) is the point of global bifurcation for associated nonhomogeneous problem. In the case N = 1 we give a description of all eigenvalues and associated eigenfunctions. Every such an eigenvalue is then the point of global bifurcation.