Stability of constant steady states of an attraction-repulsion chemotaxis system

The Cauchy problem for the attraction-repulsion chemotaxis system in the whole $n$-dimensional space has uncountable constant steady states. In the attraction chemotaxis system, each positive constant steady state is stable if it is in a certain region. On the other hand, in the repulsion chemotaxis system, every positive constant steady state is stable. Our main purpose of this paper is to give a suitable condition under which the attraction-repulsion chemotaxis system has also stable constant steady states.