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Remarks on the uniqueness problem for the logistic equation on the entire space

Published online by Cambridge University Press:  17 April 2009

Yihong Du  and
Lishan Liu
Yihong Du
Affiliation:
School of Mathematics, Statistics and Computer Science, University of New England, Armidale, NSW 2351, Australia, e-mail: ydu@turing.une.edu.au Department of Mathematics, Qufu Normal University, Qufu, Shandong, Peoples Republic of China
Lishan Liu
Affiliation:
Department of Mathematics, Qufu Normal University, Qufu, Shandong, Peoples Republic of China e-mail: lls@mail.qfnu.edu.cn
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We consider the logistic equation −Δu = a (x) u − b (x) uq on all of RN with a (x)/|x|γ and b (x)/|x|τ 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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 73 , Issue 1 , February 2006 , pp. 129 - 137
Copyright
Copyright © Australian Mathematical Society 2006

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