| Abstract: |
| We analyse positive solutions to the steady states reaction diffusion equation:
\begin{equation*}
\left\lbrace \begin{matrix} -\Delta u = \lambda \frac{1}{a}u (1-u)(a+u);~ \Omega \
\frac{\partial u}{\partial \eta}+ \gamma \sqrt{\lambda}g(u) u=0 ; ~\partial \Omega \end{matrix} \right.
\end{equation*}
where $a >0, \gamma >0,$ and $ \lambda >0$ are parameters, $\Omega$ is a bounded domain in $\mathbb{R}^N$; $N > 1$ with smooth boundary $\partial \Omega$ or $\Omega =(0,1)$, $\frac{\partial u}{\partial \eta}$ is the outward normal derivative of $u.$ In this paper, we study three emigration forms. Namely, we consider density independent emigration ($g= 1$), a negative density dependent emigration of the form $g(s) = \frac{1}{1 +\beta s}$, and a positive density dependent emigration of the form $g(s) = 1 +\beta s $, where $\beta>0$ is a parameter. We establish existence, nonexistence, and multiplicity results for ranges of $\lambda$ depending on the choice of the function $g$. We consider the case when $a\geq 1$ (logistic type growth) and the case when $0 |
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