Fully Nonlinear Logistic Equations with Sanctuary Regions and Nonlocal Diffusion
Abstract
We study positive solutions of Iu+μu=k(x)up 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 Ω0, then, as μ↑λ(Ω0), the solutions blow up locally uniformly in Ω. Writing ϕ0=ϕ(·,Ω0) for the normalized positive principal eigenfunction in Ω0, we prove uμ/|uμ|L∞(Ω)→ϕ0, uμ/|uμ|L∞(Ω)1/p →(Iϕ0/k)1/p, uniformly in ℝn and locally uniformly in Ω∖Ω0̅, respectively. Thus the upper-endpoint blow-up occurs on two distinct scales.
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BibTeX
@misc{galise2026fully,
title = {{Fully Nonlinear Logistic Equations with Sanctuary Regions and Nonlocal Diffusion}},
author = {Giulio Galise and Shuhei Kitano},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09271},
eprint = {2610.09271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09271},
}
RIS
TY - UNPB
TI - Fully Nonlinear Logistic Equations with Sanctuary Regions and Nonlocal Diffusion
AU - Giulio Galise
AU - Shuhei Kitano
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09271
UR - https://arxiv.org/abs/2610.09271
AB - We study positive solutions of Iu+μu=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 Ω_0, then, as μ↑λ(Ω_0), the solutions blow up locally uniformly in Ω. Writing ϕ_0=ϕ(·,Ω_0) for the normalized positive principal eigenfunction in Ω_0, we prove u_μ/|u_μ|_L^∞(Ω)→ϕ_0, u_μ/|u_μ|_L^∞(Ω)^1/p →(Iϕ_0/k)^1/p, uniformly in ℝ^n and locally uniformly in Ω∖Ω_0̅, respectively. Thus the upper-endpoint blow-up occurs on two distinct scales.
ER -