EXISTENCE, NONEXISTENCE AND UNIQUENESS OF POSITIVE STATIONARY SOLUTION OF A SINGULAR GIERER-MEINHARDT SYSTEM

This paper is concerned with the stationary Gierer-Meinhardt system with singularity: {d1 Delta u - a(1)u + u(p)/v(p) + p(1)(x) = 0, x is an element of Omega, d2 Delta v - a(2)v vertical bar (vr)(vs) vertical bar p(2)(x) = 0, x is an element of Omega, u(x) > 0, v(x) > 0, x is an element of Omega, u(x) = v(x) = 0, x is an element of partial derivative Omega, where -infinity < p < 1, -1 < s and q, r, d(1), d(2) are positive constants, a(1), a(2) are nonnegative constants, p(1), p(2) are smooth nonnegative functions and Omega subset of R-d (d >= 1) is a bounded smooth domain. New su ffi cient conditions, some of which are necessary, on the existence of classical solutions are established. A uniqueness result of solutions in any space dimension is also derived. Previous results are substantially improved; moreover, a much simpler mathematical approach with potential application in other problems is developed.