Accepted Paper

Inserted: 15 oct 2025
Last Updated: 15 oct 2025

Journal: Journal of Differential Equations
Volume: 452
Year: 2026
Doi: https://doi.org/10.1016/j.jde.2025.113779

Abstract:

We study a semilinear and nonlocal Neumann problem, which is the fractional analogue of the problem considered by Lin--Ni--Takagi in the '80s. The model under consideration arises in the description of stationary configurations of the Keller--Segel model for chemotaxis, when a nonlocal diffusion for the concentration of the chemical is considered. In particular, we extend to any fractional power $s\in (0,1)$ of the Laplacian (with homogeneous Neumann boundary conditions) the results obtained in 20 for $s=1/2$. We prove existence and some qualitative properties of non--constant solutions when the diffusion parameter $\varepsilon$ is small enough, and on the other hand, we show that for $\varepsilon$ large enough any solution must be necessarily constant.

Keywords: Nonlocal elliptic equations, Fractional Laplacian, Nonlocal Neumann conditions