On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type

Ni, Wei-Ming; Takagi, Izumi

doi:10.1090/S0002-9947-1986-0849484-2

On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type
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by Wei-Ming Ni and Izumi Takagi PDF

Trans. Amer. Math. Soc. 297 (1986), 351-368 Request permission

Abstract:

We derive a priori estimates for positive solutions of the Neumann problem for some semilinear elliptic systems (i.e., activator-inhibitor systems in biological pattern formation theory), as well as for semilinear single equations related to such systems. By making use of these a priori estimates, we show that under certain assumptions, there is no positive nonconstant solutions for single equations or for activator-inhibitor systems when the diffusion coefficient (of the activator, in the case of systems) is sufficiently large; we also study the existence of nonconstant solutions for specific domains.

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