Fredholm properties of radially symmetric, second order differential operators

We analyse Fredholm properties of radially symmetric second order systems in unbounded domains. The main theorem relates the Fredholm index to the Morse index at infinity. As a consequence, linear operators are Fredholm in exponentially weighted spaces for almost all weights. The result provides the basic tool for the analysis of perturbation and bifurcation problems in the presence of essential spectrum. We give a simple illustrative example, where we use the implicit function theorem to calculate the effect of a localised source term on a trimolecular chemical reaction-diffusion systems on the plane.