Steady states of Lotka-Volterra competition models with nonlinear cross-diffusion

This paper is concerned with a Lotka-Volterra model with nonlinear cross diffusion. The global existence of generalized solutions is established under some proper assumptions, and the nonexistence of nonconstant solutions is also investigated when diffusion rate is sufficiently large. At the same time, sufficient conditions ensuring the existence of non-constant solutions are obtained by using Leray-Schauder degree theory. Furthermore, steady-state bifurcation analysis is carried out in details by using Lyapunov-Schmidt reduction.