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Transactions of the American Mathematical Society

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Radially symmetric internal layers in a semilinear elliptic system


Author: Manuel A. del Pino
Journal: Trans. Amer. Math. Soc. 347 (1995), 4807-4837
MSC: Primary 35J65; Secondary 35B25, 35B60
DOI: https://doi.org/10.1090/S0002-9947-1995-1303116-3
MathSciNet review: 1303116
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Abstract: Let $ B$ denote the unit ball in $ {R^N},\quad N \geqslant 1$. We consider the problem of finding nonconstant solutions to a class of elliptic systems including the Gierer and Meinhardt model of biological pattern formation,

$\displaystyle (1.1)\qquad {\varepsilon ^2}\Delta u - u + \frac{{{u^2}}} {{1 + k{u^2}}} + p = 0\quad {\text{in}}B,$

$\displaystyle (1.2)\quad D\Delta v - v + {u^2} = 0\quad {\text{in}}B,$

$\displaystyle (1.3)\quad \frac{{\partial u}} {{\partial n}} = 0 = \frac{{\partial v}} {{\partial n}}\quad {\text{on}}\partial B,$

where $ \varepsilon $, $ D$, $ k$ and $ \rho $ denote positive constants and $ n$ the unit outer normal to $ \partial B$.

Assuming that the parameters $ \rho $, $ k$ are small and $ D$ large, we construct a family of radially symmetric solutions to (1.1)-(1.3) indexed by the parameter $ \varepsilon $, which exhibits an internal layer in $ B$, as $ \varepsilon \to 0$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1303116-3
Keywords: Elliptic system, layered solutions, radial symmetry
Article copyright: © Copyright 1995 American Mathematical Society

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