A general existence result for stationary solutions to the Keller-Segel system

TL;DR

This paper proves the existence of stationary solutions to the Keller-Segel chemotaxis model on bounded planar domains, especially non-simply connected ones, using variational and Morse-theoretical methods.

Contribution

It establishes a general existence result for stationary solutions under algebraic parameter conditions, including for non-simply connected domains.

Findings

Existence of solutions under algebraic conditions on parameters

Solutions exist for generic parameters in non-simply connected domains

Application of variational and Morse-theoretical methods

Abstract

We consider a Liouville-type PDE on a smooth bounded planar domain, which is related to stationary solutions of the Keller-Segel's model for chemotaxis. We prove existence of solutions under some algebraic conditions on the parameters. In particular, if the domain is not simply connected, then we can find solution for a generic choice of the parameters. We use variational and Morse-theoretical methods.