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

TL;DR

This paper investigates a nonlinear, nonlocal elliptic equation on a torus, demonstrating the non-uniqueness of solutions and identifying bifurcations to non-constant solutions, including explicit examples in one dimension.

Contribution

It proves the existence of non-constant solutions bifurcating from constants and constructs explicit multiple solutions in a one-dimensional case.

Findings

Constant solutions exist but are not unique.

Bifurcation analysis reveals non-constant solutions.

Explicit solutions are constructed in one dimension.

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 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.