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We focus on a system of two competing species with cross-diffusion and selfdiffusion. By constructing an appropriate auxiliary function, we derive the sufficient conditions such that there are no coexisting steady-state solutions to the model. It is worth noting that the auxiliary function constructed above is applicable to Dirichlet, Neumann and Robin boundary conditions.

Existence of steady-state solutions for a class of competing systems with cross-diffusion and self-diffusion