TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 297, Number 1, September 1986 ON THE NEUMANN PROBLEM FOR SOME SEMILINEAR ELLIPTIC EQUATIONS AND SYSTEMS OF ACTIVATOR-INHIBITOR TYPE WEI-MING NI AND IZUMI TAKAGI ABSTRACT. We derive a priori estimates for positive solutions of the Neu- mann 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 solu- tions 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. 1. Introduction. In this paper we are concerned with the Neumann problem for some semilinear elliptic equations and systems. Let flbea bounded domain in R^, tV > 2, with smooth boundary dQ, and let n denote the unit outer normal to <9fi. A typical system of equations which we consider is (1.1) dAu-u + — +<r = 0ï v ' vq ur í in fî, 1.2 DAv - vv + — = 0 (1.3) ? = ?=« ondU, dn dn where d, D, and u are positive constants, a is a nonnegative constant, N d2 3 = 1 3 is the Ar-dimensional Laplace operator, and the exponents p > 1, q > 0, r > 0, and s > 0 satisfy (1.4) 0<(p-l)/q<r/(s + l). Moreover, we are only interested in positive solutions since u and v represent the concentrations of certain substances. The system (1.1)—(1.3) was proposed by Gierer and Meinhardt [5] as a model of biological pattern formation. Stable nonconstant solutions to (1.1)—(1.3) are interpreted as the spatially inhomogeneous state of cells. Here, by the stability of solutions to elliptic equations, we mean the stability viewed as stationary solutions Received by the editors October 17, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 35J25, 35J55, 35J60. Research supported in part by NSF Grant #DMS 8200033A01. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page 351 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use