On a Fractional Lin–Ni-Takagi Type Problem with Spectral Neumann Boundary Conditions

In this seminar, we will discuss a fractional version of a semilinear Neumann problem originally studied by Lin, Ni, and Takagi in the late 1980s. This problem arises when considering the steady states of the Keller–Segel model with nonlocal diffusion of the chemical concentration. Following the approach of Stinga and Volzone, we will consider the system equipped with spectral Neumann boundary conditions. We will investigate the existence of non-constant least-energy solutions and show that, provided the diffusion parameter is small enough, these solutions attain their global maximum at exactly one point on the boundary. Furthermore, we will prove that if the diffusion parameter is sufficiently large, any solution to the system must necessarily be constant. Based on ongoing project with Eleonora Cinti (UNIBO), Marco Ghimenti (UNIPI), and Jun-cheng Wei (CUHK).