Positive steady states of an elliptic system arising from biomathematics

The existence of positive steady states of a parabolic system of reaction-diffusion equations of Keller-Segel type with chemotaxis effect and growth term arising from bacterial colony is studied. The author employs the Leray-Schauder's degree theory, a maximum principle and integral estimates to demonstrate the existence. The existence of global smooth solutions to the initial boundary value problems for the reaction-diffusion equations is also considered.