| Abstract: |
| We study existence of positive solutions of the following heterogeneous diffusive logistic equation with a harvesting term,
$$\left\{
\begin{array}{ll}
-\Delta u = \lambda a(x) u + b(x)u^2-ch(x) & \mbox{in} \,\Omega\
u(x)=0 & \mbox{on }\, \partial\Omega
\end{array}
\right.
$$
where $\Omega$ is either a bounded smooth domain or all of $\mathbb{R}^N$, in which case the boundary condition reads $\lim_{x\rightarrow\infty}u(x)=0$. Also $\lambda$ and c are positive constant, h(x), b(x) are nonnegative and there
exists a bounded smooth 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 the weighted eigenvalue problem $-\Delta v = \mu a(x)v \,\mbox{in}\, \Omega_0$ with Dirichlet boundary condition, we will show that if $h=0\, \mbox{in}\, \Omega\setminus \Omega_0,$ then our equation has a {\it unique positive solution for all c large}, provided that $\lambda$ is in a right neighborhood of $\lambda_1$. In addition we present some results on the positive solution set of this equation in the weak growth rate case complementing existing results in the literature. |
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