On a fractional semilinear Neumann problem arising in Chemotaxis

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 [23] for $s=1/2$. We prove existence and some qualitative properties of non–constant solutions when the diffusion parameter ε is small enough, and on the other hand, we show that for ε large enough any solution must be necessarily constant.