On the shape of stationary solutions to a chemotaxis model with saturation

We are concerned with stationary solutions to the chemotaxis model with effect of saturation. We study the relation of effect of saturation to the asymptotic behavior of stationary solutions. There are several saturation effects, so-called, a weak saturation, a semi-strong saturation and a strong saturation. In the weak saturation case, it is known that there exist spike solutions. However, it is unclear what types of solutions appear in the semi-strong saturation case and the strong saturation case. In this paper, in case the domain is a ball we construct a rotationally symmetric bubble-shaped solution with a finite limiting radius for the interface under the semi-strong and the strong saturation effect.