Uniqueness of boundary blow-up solutions on exterior domain of RNRN

In this paper, we consider the existence and uniqueness of positive solutions of the degenerate logistic type elliptic equation−Δu=a(x)u−b(x)|u|q−1u,x∈RN∖D,u|∂D=∞, where N⩾2N⩾2, D⊂RND⊂RN is a bounded domain with smooth boundary and a(x)a(x), b(x)b(x) are continuous functions on RNRN with b(x)⩾0b(x)⩾0, b(x)≢0b(x)≢0. We show that under rather general conditions on a(x)a(x) and b(x)b(x) for large |x||x|, there exists a unique positive solution. Our results improve the corresponding ones in [W. Dong, Y. Du, Unbounded principal eigenfunctions and the logistic equation on RNRN, Bull. Austral. Math. Soc. 67 (2003) 413–427] and [Y. Du, L. Ma, Logistic type equations on RNRN by a squeezing method involving boundary blow-up solutions, J. London Math. Soc. (2) 64 (2001) 107–124].