On the steady state problem of the chemotaxis-consumption model with logistic growth and Dirichlet boundary condition for signal

This paper concerns the steady state problem for chemotaxis consumption system with logistic growth and constant concentration of chemoattractant on the boundary of the domain. We establish the existence of a non-constant positive solution to this problem. The uniqueness of this solution is obtained under the smallness assumption on the boundary data. Some qualitative properties of the solutions and numerical results are presented.