Document Type

Article

Publication Date

4-19-2017

Publication Title

Advances in Nonlinear Analysis

Publisher

De Gruyter

Volume

8

Issue

1

First page number:

455

Last page number:

467

Abstract

We consider the equation −Δu=au−b(x)u2−ch(x) in Ω,u=0 on ∂Ω, where Ω is a smooth bounded domain in RN, b(x) and h(x) are nonnegative functions, and there exists Ω0⊂⊂Ω such that {x:b(x)=0}=Ω¯¯¯0. We investigate the existence of positive solutions of this equation for c large under the strong growth rate assumption a≥λ1(Ω0), where λ1(Ω0) is the first eigenvalue of the −Δ in Ω0 with Dirichlet boundary condition. We show that if h≡0 in Ω∖Ω¯¯¯0, then our equation has a unique positive solution for all c large, provided that a is in a right neighborhood of λ1(Ω0). For this purpose, we prove and utilize some new results on the positive solution set of this equation in the weak growth rate case.

Keywords

Logistic equation; Harvesting; Heterogeneity; Strong growth rate; Comparison principles; Stable solutions

Disciplines

Applied Mathematics

File Format

pdf

File Size

679 KB

Language

English

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