  In this paper, we prove the N-barrier maximum principle, which extends the
result in [5] from linear diffusion equations to nonlinear diffusion equations,
for a wide class of degenerate elliptic systems of porous medium type. The
N-barrier maximum principle provides a priori upper and lower bounds of the
solutions to the above-mentioned degenerate nonlinear diffusion equations
including the Shigesada-Kawasaki-Teramoto model as a special case. As an
application of the N-barrier maximum principle to a coexistence problem in
ecology, we show the nonexistence of waves in a three-species degenerate
elliptic systems.
