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

Adrian Muntean, Giulia Rui

TL;DR

The paper analyzes the elliptic problem $\nabla\cdot(a\nabla u + u\,\Phi*\nabla u)=0$ on $\mathbb{T}^d$ with nonlinear and nonlocal drift and shows that constant solutions are not unique in general. By combining spectral analysis with the Crandall--Rabinowitz bifurcation theorem, it demonstrates the existence of branches of nonconstant periodic solutions bifurcating from constant states, under a mild nondegeneracy condition on the Fourier spectrum of $\Phi$. In addition, the authors construct an explicit one-dimensional example (with $\Phi(x)=2\cos(2\pi x)$) using a modified Bessel-function ansatz to obtain an exact nontrivial solution, illustrating the bifurcation and non-uniqueness phenomenon concretely. These results illuminate inherent non-uniqueness in periodic homogenization problems with nonlocal drifts and provide both qualitative bifurcation insight and a concrete 1D construction. The work connects spectral properties of the kernel $\Phi$ to the multiplicity of solutions and highlights the role of nonlocal effects in elliptic problems on periodic domains.

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.

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

TL;DR

The paper analyzes the elliptic problem on with nonlinear and nonlocal drift and shows that constant solutions are not unique in general. By combining spectral analysis with the Crandall--Rabinowitz bifurcation theorem, it demonstrates the existence of branches of nonconstant periodic solutions bifurcating from constant states, under a mild nondegeneracy condition on the Fourier spectrum of . In addition, the authors construct an explicit one-dimensional example (with ) using a modified Bessel-function ansatz to obtain an exact nontrivial solution, illustrating the bifurcation and non-uniqueness phenomenon concretely. These results illuminate inherent non-uniqueness in periodic homogenization problems with nonlocal drifts and provide both qualitative bifurcation insight and a concrete 1D construction. The work connects spectral properties of the kernel to the multiplicity of solutions and highlights the role of nonlocal effects in elliptic problems on periodic domains.

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.
Paper Structure (6 sections, 2 theorems, 56 equations)

This paper contains 6 sections, 2 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. An observation and the statement of our main result
  3. Observation
  4. Statement of the main result
  5. Proof of Theorem \ref{['thm:nonuniqueness']}
  6. A one-dimensional example with explicit construction of a non-trivial solution

Key Result

Theorem 2.1

Assume that there exists an index $k_0\in\mathbb{Z}^d\setminus\{0\}$ such that: Then there exists $\varepsilon>0$ and two smooth curves $\varphi, \psi$ such that for a suitable $v_0 \in \mathbb{H}^{s+2}(\Omega)$, where $s\geq d/2$. Specifically, there exists a smooth curve of nonconstant solutions to eq:main.

Theorems & Definitions (4)

  • Theorem 2.1: Non-uniqueness of periodic solutions to $(\star)$
  • Theorem 3.1: Crandall--Rabinowitz
  • Remark 1
  • Remark 2