Nondegeneracy of the second bifurcating branches for the Chafee-Infante problem on a planar symmetric domain

Abstract

Let Ω be a planar domain such that Ω is symmetric with respect to both the x- and y-axes and Ω satisfies certain conditions. Then the second eigenvalue of the Dirichlet Laplacian on Ω, υ 2 (Ω), is simple, and the corresponding eigenfunction is odd with respect to the y-axis. Let f ∈ C 3 be a function such that > 0, f'(0) 0. Let C denote the maximal continua consisting of nontrivial solutions, {(λ, u)}, to Δu + Δf(u) = 0 in Ω, u = 0 on ∂Ω and emanating from the second eigenvalue (v 2 (Ω)/f'(0),0). We show that, for each (λ, u) ∈ C, the Morse index of u is one and zero is not an eigenvalue of the linearized problem. We show that C consists of two unbounded curves, each curve is parametrized by λ and the closure C is homeomorphic to ℝ.