On the spectrum of some linear noncooperative elliptic systems with radial symmetry

Abstract

In this work we study the spectrum of some linear weakly coupled noncooperative elliptic systems with two species subject to homogeneous Dirichlet boundary conditions. When the domain supporting the species is a ball or an annulus we prove that zero is never an eigenvalue. As a consequence of this result we also show that the classical Lotka-Volterra predator-prey model with diffusion and radial function coefficients has a unique component-wise radial coexistence state with no zero eigenvalue of the linearization on the space $W^{2,p}(D)$, $p > {N\over 2}$. Therefore, a radially symmetric coexistence state only may lose stability by a Hopf bifurcation when we vary parameters.