Nondegeneracy of the Second Bifurcating Branches for the Chafee-infante Problem on a Planar Symmetric Domain

Let Ω be a planar domain such that Ω is symmetric with respect to both the xand y-axes and Ω satisfies certain conditions. Then the second eigenvalue of the Dirichlet Laplacian on Ω, ν2(Ω), is simple, and the corresponding eigenfunction is odd with respect to the y-axis. Let f ∈ C3 be a function such that f ′(0) > 0, f ′′′(0) < 0, f(−u) = −f(u) and d du ( f(u) u ) < 0 for u > 0. Let C denote the maximal continua consisting of nontrivial solutions, {(λ, u)}, to Δu+ λf(u) = 0 in Ω, u = 0 on ∂Ω and emanating from the second eigenvalue (ν2(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.