Sharp uniqueness threshold and minimal solution regularity for ordinary differential equations with singular coefficients

For the linear-in-y n-th order ordinary differential equation $y^{(n)}+\sum_{k=0}^{n-1}a_k(x,y)y^{(k)}=0$ on an interval containing $0$, Pan and Wang proved that any $C^\infty$ solution with zero $(n-1)$-jet at the origin vanishes identically near $0$ whenever the singularity sizes satisfy $\limsup_{x\to0}|x|^{n-k}|a_k(x,y)|\le 1/e$, relaxable for fixed $n\ge 2$ to $\limsup<1/B_n$ with $B_n=\sum_{k=0}^{n-1}1/k!$. Their examples show only that some finite bound is necessary (a Bessel-type example with integer parameter $m\ge 2$ fails at $m^2\ge 4$) and that at low regularity ($C^{1,\alpha}$ solutions) no bound of this type can give uniqueness. The derived open problem is to determine the exact uniqueness boundary: the optimal constant $C_n$ (the sharp threshold for the common limsup bound) for each $n\ge2$, and the minimal solution regularity under which uniqueness holds at a given coefficient-size level. No later work determines either quantity.

Background

Classical uniqueness theory for ordinary differential equations (the Picard–Lindelöf and related criteria) assumes coefficients that are continuous, and typically Lipschitz in the unknown. Here one considers the opposite situation: the n-th order equation

$y^{(n)}+a_{n-1}(x,y)y^{(n-1)}+\cdots+a_0(x,y)y=0,\qquad x\in(-a,a),\ a>0,$

which is linear in $y$ but whose coefficients $a_k(x,y)$ may depend arbitrarily on $(x,y)$ and may blow up at the isolated singular point $x=0$. No continuity or measurability assumption is made beyond a size condition: the natural scaling compares $|a_k(x,y)|$ with $|x|^{-(n-k)}$, the strength at which the k-th derivative term becomes borderline, via the quantities $\limsup_{x\to0}|x|^{n-k}|a_k(x,y)|$. The uniqueness question studied is local unique continuation from a singular point: if a solution has zero initial jet $f(0)=f'(0)=\cdots=f^{(n-1)}(0)=0$, must it vanish on a neighborhood of $0$?

Yifei Pan and Mei Wang (arXiv:0812.3427; published as Electron. J. Differ. Equ. 2009, No. 56) proved that the answer is yes for $C^\infty$ solutions when all the limsups are at most $1/e$ (Theorem 1), and, for each fixed $n$, when they are strictly below $1/B_n$, where $B_n=\sum_{k=0}^{n-1}1/k!$ (a remark in the same paper; note $1/B_n>1/e$ for every finite $n\ge2$, so this is the stronger sufficient bound). Their method reduces the equation to a differential inequality $|f^{(n)}|\le C\sum_{k=0}^{n-1}|f^{(k)}|/|x|^{n-k}$ and applies their earlier companion result on non-flat functions (a Carleman-type unique-continuation lemma: a $C^\infty$ function satisfying such an inequality that is flat at $0$ — all of its derivatives vanish there — is identically zero, for $n\ge2$). The operative criterion (their Proposition 9, numbered Proposition 2.3 in the published version) is sharper than the stated theorems: uniqueness holds whenever $f^{(k)}(0)=0$ for all $k\le\lfloor B_nC+n-1\rfloor$, where $C=\max_k\limsup_{x\to0}|x|^{n-k}|a_k(x,y)|$ is the coefficient-size datum of the given equation; the $C^\infty$ hypothesis is used because this derivative order generally exceeds $n$.

Two of their examples delimit the question. A Bessel-equation example ($n=2$; the classical equation $x^2y''+xy'+(x^2-m^2)y=0$ for Bessel functions) gives a nonzero $C^\infty$ solution with zero initial jet when $\limsup|x|^2|a_0|=m^2\ge4$ for an integer $m\ge2$, showing some finite size bound is necessary. A second example shows that for solutions of only Hölder class $C^{1,\alpha}$ (first derivative Hölder continuous of order $\alpha$) non-uniqueness occurs with arbitrarily small limsup constants, so uniqueness genuinely depends on solution regularity, not on the size bound alone. A third family, the Cauchy–Euler (equidimensional) equations with constant coefficients and power-type solutions, sits between these categories and hints that the true threshold lies in the gap left open by the theorems. For first-order equations ($n=1$) the corresponding questions belong to the classical Osgood-type uniqueness criteria (and the Euler example $y'=y/x$ already shows a bound of size $1$ is necessary), but the method and results of the cited work require $n\ge2$, which is the range of interest here.

Problem Statement

For each integer $n\ge2$, consider the equation

$y^{(n)}+a_{n-1}(x,y)y^{(n-1)}+\cdots+a_0(x,y)y=0,\qquad x\in(-a,a),$

with coefficients $a_k(x,y)$ subject only to the size condition $\limsup_{x\to0}|x|^{n-k}|a_k(x,y)|\le C$ for $k=0,1,\ldots,n-1$, evaluated — following the convention of the source's proofs, which consume the coefficients only along the given solution — along $y=f(x)$, i.e. as $\limsup_{x\to0}|x|^{n-k}|a_k(x,f(x))|\le C$; for coefficients depending solely on $x$ (as in all examples cited below) this coincides with a uniform-in-$y$ reading.

(1) Determine the optimal constant $C_n$: the sharp threshold, i.e. the supremum of the values $C$ for which every $C^\infty$ solution with zero initial jet $f(0)=f'(0)=\cdots=f^{(n-1)}(0)=0$ must vanish on some neighborhood $[-\delta,\delta]$ of the singular point. Known: $C_n\ge1/B_n$ with $B_n=\sum_{k=0}^{n-1}1/k!$ (uniqueness holds for limsups strictly below $1/B_n$; $1/e$ is the corresponding uniform-in-$n$ bound), while the source's examples show only that finiteness of some bound is necessary — the Bessel example fails at $m^2\ge4$ (integer $m\ge2$, $n=2$), and elementary constant-coefficient substitutions in the paper's Cauchy–Euler family (its Example 6; for instance $y=x^2$ with the scaled coefficients $a_1(x)=-2/(3x)$, $a_0(x)=-2/(3x^2)$, giving non-uniqueness at the common bound $C=2/3>1/B_2=1/2$, and analogously $y=x^n$ with $a_k(x)=-c/x^{n-k}$, $c=1/\sum_{j=1}^n1/j!$, for general $n$) exhibit non-uniqueness at constants strictly larger than $1/B_n$, though the authors themselves compute no critical value — leaving the exact value of $C_n$ undetermined.

(2) Determine the minimal solution regularity that preserves uniqueness under such coefficient-size conditions: the source proves uniqueness for $C^\infty$ solutions (its proof needs derivatives of order up to $\lfloor B_nC+n-1\rfloor$, which can exceed $n$) and exhibits non-uniqueness for $C^{1,\alpha}$ solutions at arbitrarily small constants; characterize what holds in between (for each of the three regularity scales considered here — integer classes $C^k$, Hölder classes $C^{k,\alpha}$, and Sobolev classes $W^{k,p}$ — whether uniqueness holds and how the corresponding threshold depends on the size constant; the minimal regularity sought is the weakest class at which uniqueness holds).

These two deliverables are two coordinates of one objective: the exact uniqueness boundary, in coefficient size and in solution regularity, for linear ordinary differential equations with power-type singular coefficients at an isolated point. Extensions to non-powerlike or oscillatory singular behavior, while a natural outlook, are not part of this problem as the source supports no determinate formulation of them.

References

1. Yifei Pan, Mei Wang, "A uniqueness result on ordinary differential equations with singular coefficients", arXiv:0812.3427 (2008), https://arxiv.org/abs/0812.3427
2. Yifei Pan, Mei Wang, "A uniqueness result for ordinary differential equations with singular coefficients", Electronic Journal of Differential Equations, Vol. 2009 (2009), No. 56, pp. 1–6, Zbl 1170.34306, https://ejde.math.txstate.edu/Volumes/2009/56/pan.pdf
3. Yifei Pan, Mei Wang, "When is a function not flat?", Journal of Mathematical Analysis and Applications 340 (2008), no. 1, 536–542, DOI 10.1016/j.jmaa.2007.08.044, https://doi.org/10.1016/j.jmaa.2007.08.044 (Zbl 1131.26016)
4. Yifei Pan, Mei Wang, "Uniqueness for n-th order differential systems with strong singularities", Electronic Journal of Differential Equations, Vol. 2010 (2010), No. 172, pp. 1–9, Zbl 1227.34014, https://ejde.math.txstate.edu/Volumes/2010/172/pan.pdf
5. Yifei Pan, Mei Wang, Yu Yan, "Uniqueness theorem for ordinary differential equations with Hölder continuity", Pacific Journal of Mathematics 263 (2013), no. 2, 453–473, DOI 10.2140/pjm.2013.263.453, https://doi.org/10.2140/pjm.2013.263.453; preprint arXiv:1209.6064, https://arxiv.org/abs/1209.6064
6. Nuno Costa Dias, Cristina Jorge, João Nuno Prata, "Ordinary differential equations with singular coefficients: an intrinsic formulation with applications to the Euler–Bernoulli beam equation", Journal of Dynamics and Differential Equations 33 (2021), no. 2, 593–619, DOI 10.1007/s10884-020-09822-x, https://doi.org/10.1007/s10884-020-09822-x, Zbl 1468.34018; preprint arXiv:2001.10141, https://arxiv.org/abs/2001.10141
7. Yifei Pan, Yu Yan, Yuan Zhang, "On a Local Radial Rigidity Phenomenon of Elliptic Systems on Riemannian Manifolds", arXiv:2606.00554 (2026), https://arxiv.org/abs/2606.00554