A heterogeneous diffusive logistic model with constant yield harvesting in $\mathbb{R}^N$ under strong growth rate

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.