In this seminar, we will explore a fractional version of a semilinear Neumann problem studied by Lin–Ni–Takagi in the late 1980s. The problem arises when considering steady states of the Keller–Segel model with nonlocal diffusion of chemical concentration. In the fractional setting, there are different possible notions of 'nonlocal Neumann conditions'. We will study the system with both spectral conditions, following Stinga and Volzone, and integral conditions (introduced by Dipierro, Ros-Oton, and Valdinoci), as investigated by Cinti and Colasuonno. While an extension theorem analogous to that of Caffarelli and Silvestre exists for the spectral boundary condition, no such theorem is available for integral boundary conditions. This makes it more challenging to obtain Liouville-type theorems and to prove the nonexistence of non-trivial positive solutions when the concentration parameter is sufficiently large.

Based on joint projects with Eleonora Cinti (UNIBO), Marco Ghimenti (UNIPI), Juncheng Wei (CUHK), and Tobias Weth (GUF).