  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.
