Uniqueness and existence of positive solutions for degenerate logistic type elliptic equations on [formula omitted]

Abstract

In this paper, we consider the existence and uniqueness of positive solutions for the degenerate logistic type equation − Δ u = a ( x ) u − b ( x ) u q , x ∈ R N ( N ≥ 2 ) . We show that under rather general conditions on a ( x ) and b ( x ) , there exists a unique positive solution for large | x | . Our results are an improvement on the corresponding ones in [W. Dong, Y. Du, Unbounded principal eigenfunctions and the logistic equation on R N , Bull. Austral. Math. Soc. 67 (2003) 413–427] and [Y. Du, L. Ma, Logistic type equations on R N by a squeezing method involving boundary blow-up solutions, J. London Math. Soc. 64 (2) (2001) 107–124].