Uniqueness of boundary blow-up solutions on unbounded domain of RN

In this paper, we consider the existence of positive solutions of the degenerate logistic type elliptic equation -Delta u = a(x)u - b(x)f(u), x is an element of R-N\ Du vertical bar(partial derivative D) = infinity, where N >= 2, D subset of R-N is a bounded smooth domain and a(x), b(x) are continuous functions on R-N with b(x) >= 0, b(x) not equivalent to 0, especially b(x) = 0 on partial derivative D and f(u) is an element of C(0, infinity). We show that under rather general conditions on a(x) and b(x) for large vertical bar x vertical bar and f(u) behaves like u(q) where q > 1. Without the behavior of b(x) near the boundary partial derivative D, it will be shown that there exists a unique positive solution. Our results improve the corresponding ones in [W. Dong, Pang, Uniqueness of boundary blow-up solutions on exterior domain of R-N, J. Math. Anal. Appl. 330 (2007), 654-664] and [Y. Du,L. Ma, Logistic type equations on R-N by a squeezing method involving boundary blow-up solutions, J. London Math. Soc., (2) 64 (2001), 107-124]. (C) 2009 Elsevier Ltd. All rights reserved.