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

Luca Battaglia

doi:10.3934/dcds.2019038

Article Contents

2019, Volume 39, Issue 2: 905-926. Doi: 10.3934/dcds.2019038

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

Luca Battaglia

Università degli Studi Roma Tre, Dipartimento di Matematica e Fisica, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy

Received: January 31, 2018

Revised: July 31, 2018

Published: January 2019

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Abstract

We consider the following Liouville-type PDE, which is related to stationary solutions of the Keller-Segel's model for chemotaxis:

$\left\{ \begin{gathered} - \Delta u + \beta u = \rho \left( {\frac{{{e^u}}}{{\int_\Omega {{e^u}} }} - \frac{1}{{\left| \Omega \right|}}} \right)\;\;\;\;\;\;{\text{in}}\;\Omega \hfill \\ {\partial _\nu }u = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{on}}\;\partial \Omega \hfill \\ \end{gathered} \right.,$

where $\Omega \subset {\mathbb{R}^2}$ is a smooth bounded domain and $\beta, ρ$ are real parameters. We prove existence of solutions under some algebraic conditions involving $\beta, ρ$. In particular, if $\Omega$ is not simply connected, then we can find solution for a generic choice of the parameters. We use variational and Morse-theoretical methods.

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Mathematics Subject Classification: Primary: 35J61; Secondary: 35J20.

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