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.
Citation
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
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