Studying the Fisher-KPP evolution equation I came across the steady state elliptic problem which can be written in the following form:
$$ \begin{cases} -d\Delta Y(x)=r(x)Y(x)\left (1-\dfrac{Y(x)}{K(x)}\right ),\ x\in\Omega \\ \dfrac{\partial Y}{\partial \nu}=0, \ x\in\partial\Omega\end{cases} $$
Hereabove $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain and $K\in L^{\infty}(\Omega)$ is the carrying capacity. $r=r(x)$ is a function from $L^{\infty}(\Omega)$ and $d>0$ is a positive constant. We assume that there are two constants $k_0, r_0>0$ such that $K(x)>k_0$ and $r(x)>r_0$ for almost all $x\in\Omega$.
Since for the logistic equation from ODE $y'(t)=ry(t)\left (1-\dfrac{y(t)}{k}\right )$ the stationary solution $y(t)=k>0$ for all $t\geq 0$ is the unique strict positive and globally asimptotically stable I think the for the spatial model this will hold too. So presumably there will be a unique strict positive solution $Y$ for the elliptic problem which is globally asimptotically stable. I cannot prove this. For any advice I will be thankful.
I know that some monotone techniques will be useful here but I cannot see how. I tried to use upper and lower solutions for this problem and some eigenvalue problems but I did not succeed in proving anything relevant.
