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2022 A heterogeneous diffusive logistic model with constant yield harvesting in $\mathbb{R}^N$ under strong growth rate
Hossein Tehrani
Topol. Methods Nonlinear Anal. 59(1): 385-408 (2022). DOI: 10.12775/TMNA.2021.034

Abstract

We study existence of positive solutions of the following heterogeneous diffusive logistic equation with a harvesting term,\begin{equation*}-\Delta u =\lambda a(x) u -b(x) u^2 - c h(x), \quad\text{in } \mathbb{R}^N,\qquad \lim_{|x|\rightarrow\infty}u(x)=0,\end{equation*}where $\lambda$ and $c$ are positive constant, $h(x)$, $b(x)$ are nonnegative and there exists a bounded region $\Omega_0$ such that $\overline{\Omega}_0 = \{ x : b(x)=0 \}$. Under the strong growth rate assumption, that is, when $\lambda \geq \lambda_1(\Omega_0)$, the first eigenvalue of weighted eigenvalue problem $-\Delta v=\mu a(x)v$ in $\Omega_0$ with Dirichlet boundary condition, we will show that if $h \equiv 0$ in $\mathbb{R}^N\setminus\overline{\Omega}_0$ then our equation has a unique positive solution for all $c$ large, provided that $\lambda$ is in a right neighborhood of $\lambda_1 (\Omega_0)$. In addition we prove a new result on the positive solution set of this equation in the weak growth rate case complimenting existing results in the literature.

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Hossein Tehrani. "A heterogeneous diffusive logistic model with constant yield harvesting in $\mathbb{R}^N$ under strong growth rate." Topol. Methods Nonlinear Anal. 59 (1) 385 - 408, 2022. https://doi.org/10.12775/TMNA.2021.034

Information

Published: 2022
First available in Project Euclid: 4 April 2022

Digital Object Identifier: 10.12775/TMNA.2021.034

Keywords: Diffusive logistic equation , harvesting term , strong growth rate , whole space $\mathbb R^N$

Rights: Copyright © 2022 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.59 • No. 1 • 2022
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