Remarks on the uniqueness problem for the logistic equation on the entire space
Du, Yihong; Liu, Lishan; Du Yihong; School of Mathematics, Statistics and Computer Science, University of New England; Qufu Normal University; Liu Lishan; Qufu Normal University
Журнал:
Bulletin of the Australian Mathematical Society
Дата:
2006
Аннотация:
We consider the logistic equation −Δu = a (x) u − b (x) u<sup>q</sup> on all of R<sup>N</sup> with a (x)/|x|<sup>γ</sup> and b (x)/|x|<sup>τ</sup> bounded away from 0 and infinity for all large |x|, where γ > −2, τ ∈ (−∞, ∞). We show that this problem has a unique positive solution. This considerably improves some earlier results. The main new technique here is a Safonov type iteration argument. The result can also be proved by a technique introduced by Marcus and Veron, and the two different techniques are compared.
335.8Кб