Advances in Differential Equations

Positive solutions for singular semilinear elliptic systems

Jesús Hernández, Francisco J. Mancebo, and José M. Vega

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We establish existence results for singular semilinear elliptic systems on bounded domains with homogeneous Dirichlet boundary conditions. The systems considered are the paradigmatic mathematical models of chemical reactions, morphogenesis (singular Gierer-Meinhardt system) and population dynamics. In these systems the operator need not be in divergence form and the systems need not be cooperative. The results have been obtained by the method of sub and supersolutions (appropriately modified) and Schauder's fixed point theorem. Some uniqueness results have been obtained extending a "concavity" argument used for a single equation. We extend some existence results to general elliptic operators and more general nonlinearities and we prove existence for systems that have not been considered in the literature.

Article information

Adv. Differential Equations Volume 13, Number 9-10 (2008), 857-880.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J57: Boundary value problems for second-order elliptic systems
Secondary: 35A01: Existence problems: global existence, local existence, non-existence 35J61: Semilinear elliptic equations 35J75: Singular elliptic equations


Hernández, Jesús; Mancebo, Francisco J.; Vega, José M. Positive solutions for singular semilinear elliptic systems. Adv. Differential Equations 13 (2008), no. 9-10, 857--880.

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