Fully Nonlinear Logistic Equations with Sanctuary Regions and Nonlocal Diffusion

We study positive solutions of $Iu+\mu u=k(x)u^p$ in a bounded domain, where $p>1$, the nonnegative coefficient $k$ may vanish on a sanctuary region, and $I$ is a fully nonlinear nonlocal operator uniformly elliptic in the Caffarelli–Silvestre sense. We characterize the sharp existence interval in terms of positive principal half-eigenvalues and prove uniqueness. At the lower endpoint, the solutions vanish uniformly and their normalized profiles converge to the principal eigenfunction. If $k$ vanishes on a sanctuary ${\mathrm{\Omega }}_0$, then, as $\mu \uparrow \lambda ({\mathrm{\Omega }}_0)$, the solutions blow up locally uniformly in $\mathrm{\Omega }$. Writing ${\phi }_0=\phi (\cdot ,{\mathrm{\Omega }}_0)$ for the normalized positive principal eigenfunction in ${\mathrm{\Omega }}_0$, we prove $$\frac{u_{\mu }}{\mathrm{\parallel }{u}_{\mu }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}}\to {\phi }_0,\frac{u_{\mu }}{\mathrm{\parallel }{u}_{\mu }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}^{1\mathrm{/}p}}\to {\left(\frac{I{\phi }_0}{k}\right)}^{1\mathrm{/}p},$$ uniformly in ${\mathbb{R}}^n$ and locally uniformly in $\mathrm{\Omega }\setminus \overline{{\mathrm{\Omega }}_0}$, respectively. Thus the upper-endpoint blow-up occurs on two distinct scales.