Fixed points of maps and nontrivial weak solutions to a class of nonlinear strongly coupled elliptic systems

Local indices at isolated fixed points of a differentiable compact nonlinear map $T$ on Banach spaces will be discussed. These results are applied to establish the existence of nontrivial solutions. As an example, the existence of nontrivial weak solutions to a class of nonlinear strongly coupled elliptic systems of two equations will also be studied. A bifurcation phenomenon due to cross-diffusion is reported.