Uniqueness of boundary blow-up solutions on unbounded domain of image Original Research Article

In this paper, we consider the existence of positive solutions of the degenerate logistic type elliptic equation View the MathML source−Δu=a(x)u−b(x)f(u),x∈RN∖Du|∂D=∞, Turn MathJax on where N≥2N≥2, D⊂RND⊂RN is a bounded smooth domain 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, especially b(x)=0b(x)=0 on ∂D∂D and f(u)∈C(0,∞)f(u)∈C(0,∞). We show that under rather general conditions on a(x)a(x) and b(x)b(x) for large |x||x| and f(u)f(u) behaves like uquq where q>1q>1. Without the behavior of b(x)b(x) near the boundary ∂D∂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 RNRN, J. Math. Anal. Appl. 330 (2007), 654–664] 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].