| Popis: |
In this paper, we consider the existence and uniqueness of positive solutions of the degenerate logistic type elliptic equation − Δ u = a ( x ) u − b ( x ) | u | q − 1 u , x ∈ R N ∖ D , u | ∂ D = ∞ , where N ⩾ 2 , D ⊂ R N is a bounded domain with smooth boundary and a ( x ) , b ( x ) are continuous functions on R N with b ( x ) ⩾ 0 , b ( x ) ≢ 0 . We show that under rather general conditions on a ( x ) and b ( x ) for large | 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 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. (2) 64 (2001) 107–124]. |
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