A bifurcation result for a Keller-Segel-type problem

We consider a parametric elliptic problem governed by the spectral Neumann fractional Laplacian on a bounded domain of with a general nonlinearity. This problem is related to the existence of steady states for Keller-Segel systems in which the diffusion of the chemical is nonlocal. By variational arguments we prove the existence of a weak solution as a local minimum of the corresponding energy functional and we derive some qualitative properties of this solution. Finally, we prove a regularity result for weak solutions of the problem under consideration, which is of independent interest.