Large solutions for an elliptic system of quasilinear equations

In this paper we consider the quasilinear elliptic system Delta(p)u = u(a)v(b), Delta(p)v = u(c)v(e) in a smooth bounded domain Omega subset of R-N, with the boundary conditions u = v = +infinity on partial derivative Omega. The operator Ap stands for the p-Laplacian defined by Delta(p)u = div(vertical bar del u vertical bar(p-2)del u), p > 1, and the exponents verify a, e > p - 1, b, c > 0 and (a - p + 1)(e - p + 1) >= bc. We analyze positive solutions in both components, providing necessary and sufficient conditions for existence. We also prove uniqueness of positive solutions in the case (a - p + 1)(e - p + 1) > bc and obtain the exact blow-up rate near the boundary of the solution. In the case (a - p + 1)(e - p + 1) = bc, infinitely many positive solutions are constructed. (c) 2008 Elsevier Inc. All rights reserved.