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

2026/02/02 by Adrian Muntean, Giulia Rui
#math.AP

paper · pdf

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.

Citations

• Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55)

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