Lack of uniqueness for an elliptic equation with nonlinear and nonlocal drift posed on a torus

We study a nonlinear and nonlocal elliptic equation posed on the flat torus.

While constant solutions always exist, we show that uniqueness fails in general.

Using spectral analysis and the Crandall--Rabinowitz bifurcation theorem, we prove the existence of branches of non-constant periodic solutions bifurcating from constant states.

This result is qualitative and non-constructive.

Using a conceptually different argument, we construct explicit multiple solutions for a specific one--dimensional formulation of our target problem.