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

Abstract

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. Copyright Clearance Centre, Inc.