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A singular Gierer-Meinhardt system of elliptic equations: the classical case. (English) Zbl 1140.35408
Summary: This paper studies the existence of solutions of a nonlinear elliptic system usually called the generalized Gierer-Meinhardt equations \[ \begin{cases} d_ {1}\Delta u-\alpha_ {1}u+\gamma_ {1}\frac{u^ {r}}{v}=0,\\ d_ {2}\Delta v-\alpha_ {2}v+\gamma_ {2}u^ {r}=0,\\ u| _ {\partial\Omega}=0,v| _ {\partial\Omega}=0.\\ \end{cases} \] This type of system originally arose in studies of pattern-formation in biology, and has interesting and challenging mathematical properties, especially with Dirichlet boundary conditions when the nonlinear terms become singular near the boundary. We study the existence of solutions in the classical case of Gierer-Meinhardt [A. Gierer and H. Meinhardt, Kybernetik 12, No. 1, 30–39 (1972)], when the activator-inhibitor model has different sources.

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35J60 Nonlinear elliptic equations
Full Text: DOI
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