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Existence of weak solutions for some elliptic systems

Lucio Boccardo and Luigi Orsina
Vol. 4 (2022), No. 3, 517–534
DOI: 10.2140/paa.2022.4.517
Abstract

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

{ −Δu + u = −Adiv ⁡ (u ∇ ⁡ψ 1+ψ) + f(x)  in Ω, −Δψ = uλ   in Ω.

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 < 1 2.

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