Structure on the Set of Stationary Solutions for a Two Competing Species Model with Density-Dependent Diffusion

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In this research, we study the structure on the set of radially symmetric positive stationary solutions for a two competing species model with density-dependent diffusion, when the habitat of the community is the inside of a ball. Although the dimension of the habitat is an integer, we assume that it can be any real number. To establish the structure, we focus on the case where the diffusion rate of the species is positive and sufficiently small, employ the bifurcation theory and the comparison principle, and then investigate the property of eigenvalues and their corresponding eigenfunctions for the linearized operator around the stationary solution.