Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems

Abstract

The main result of this work is a Dancer-type bifurcation result for the quasilinear elliptic problem { - Δp u = λ u p-2u + h (x,u(x); λ in Ω u = 0 on ∂ Ω. Here, Ω is a bounded domain in ℝN (N ≥ 1), Δp u = div ( u p-2 u) denotes the Dirichlet p-Laplacian on W1,p0(Ω), 1 < p 2) in an interesting particular case. Our proofs are based on very precise, local asymptotic analysis for λ near μ1 (for any 1 < p < ∞) which is combined with standard topological degree arguments from global bifurcation theory used in Dancer's original work. © 2008 Birkhaueser.