Uniqueness and existence of positive solutions for degenerate logistic type elliptic equations on RNRN

In this paper, we consider the existence and uniqueness of positive solutions for the degenerate logistic type equation−Δu=a(x)u−b(x)uq,x∈RN(N≥2). We show that under rather general conditions on a(x)a(x) and b(x)b(x), there exists a unique positive solution for large |x||x|. Our results are an improvement on 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. 64 (2) (2001) 107–124].