Existence of weak solutions for some elliptic systems

We study existence of positive weak solutions for stationary systems weakly related to the logarithmic Keller–Segel model for chemotaxis. The simplest case is the Dirichlet problem for

{\begin{matrix} - Δ u + u = - A div (u \frac{\nabla ψ}{1 + ψ}) + f (x) & in Ω, \\ - Δ ψ = u^{λ} & in Ω . \end{matrix}

We prove existence results under conditions on $A > 0$ and $λ > 0$ ; for instance, we prove existence of finite energy solutions $u$ if $0 < A < \frac{1}{2}$ .

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Keywords

elliptic systems, Keller–Segel model, nonlinear elliptic equations

Mathematical Subject Classification

Primary: 35J47, 35J60

Milestones

Received: 9 July 2021

Revised: 27 December 2021

Accepted: 22 March 2022

Published: 4 December 2022

Authors

Lucio Boccardo
Dipartimento di Matematica Sapienza Università di Roma Rome Italy

Luigi Orsina
Dipartimento di Matematica Sapienza Università di Roma Rome Italy

Vol. 4, No. 3, 2022

Lucio Boccardo and Luigi Orsina

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors