Linearized stability implies asymptotic stability for radially symmetric equilibria of p-Laplacian boundary value problems in the unit ball in ℝ^N

We consider the parabolic initial-boundary value problem (Forumala presented). where Ω = B ₁ is the unit ball centered at the origin in R ^N, with N ≥ 2, p > 1, and ∆ _p denotes the p-Laplacian on Ω. The function f: [0, 1] × R → R is continuous, and the partial derivative f _v exists and is continuous and bounded on [0, 1] × R. It will be shown that (under certain additional hypotheses) the ‘principle of linearized stability’ holds for radially symmetric equilibrium solutions u ₀ of the equation. That is, the asymptotic stability, or instability, of u ₀ is determined by the sign of the principal eigenvalue of a linearization of the problem at u ₀. It is well-known that this principle holds for the semilinear case p = 2 (∆ ₂ is the linear Laplacian), but has not been shown to hold when p ≠ 2. We also consider a bifurcation type problem similar to the one above, having a line of trivial solutions and a curve of non-trivial solutions bifurcating from the line of trivial solutions at the principal eigenvalue of the p-Laplacian. We characterize the stability, or instability, of both the trivial solutions and the non-trivial bifurcating solutions, in a neighbourhood of the bifurcation point, and we obtain a result on ‘exchange of stability’ at the bifurcation point, analogous to the well-known result when p = 2.