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Existence of solution for a class of activator–inhibitor systems

Part of: Partial differential equations Elliptic equations and systems Physiological, cellular and medical topics

Published online by Cambridge University Press:  12 April 2022

Giovany Figueiredo Open the ORCID record for Giovany Figueiredo [Opens in a new window]  and
Marcelo Montenegro
Giovany Figueiredo
Affiliation:
Departamento de Matemática, Universidade de Brasília, Campus Darcy Ribeiro, 01, Brasília, DF, Brazil E-mail: [email protected]
Marcelo Montenegro
Affiliation:
Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Rua Sérgio Buarque de Holanda, 651, Campinas, SP, Brazil E-mail: [email protected]
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Abstract

We prove the existence of a solution for a class of activator–inhibitor system of type $- \Delta u +u = f(u) -v$ , $-\Delta v+ v=u$ in $\mathbb{R}^{N}$ . The function f is a general nonlinearity which can grow polynomially in dimension $N\geq 3$ or exponentiallly if $N=2$ . We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.

MSC classification

Primary: 35A15: Variational methods 35B33: Critical exponents 35J47: Second-order elliptic systems 92C20: Neural biology
Secondary: 35J50: Variational methods for elliptic systems
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 65 , Issue 1 , January 2023 , pp. 98 - 113
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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