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

Abstract

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 nonconstant periodic solutions bifurcating from constant states. This result is qualitative and nonconstructive. Using a conceptually different argument, we construct explicit multiple solutions for a specific one-dimensional formulation of our target problem.