Let T>0, let a:[0,T]→R be a two-step indefinite weight, and let gamma∈Rsetminus-1,0,1. The conjecture asserts that the boundary-value problem u''+a(t)u^gamma=0, with either Neumann conditions u'(0)=u'(T)=0 or periodic conditions u(0)=u(T) and u'(0)=u'(T), has at most one positive solution u(t)>0 on [0,T].

Boscaggin–Feltrin–Zanolin uniqueness conjecture

The conjecture concerns uniqueness and existence of positive solutions for two-step indefinite $p$ -Laplacian boundary-value problems with Neumann or periodic conditions. Boscaggin, Feltrin, and Zanolin formulated the relevant classification for power nonlinearities.

Known results

For the classical Laplacian, uniqueness was established for $\gamma\in(-\infty,-3]\cup(1,\infty)$ (Boscaggin, Feltrin, and Zanolin, 2020).
In that range, existence is characterized by $\gamma\int_0^T a(t)\,\mathrm{d}t<0$ .
For general $p>1$ , the claimed range is $\gamma\in\left(-\infty,\frac{1-2p}{p-1}\right]\cup(p-1,\infty)$ .
For $-1<\gamma<0$ , the earlier work reported existence but no general uniqueness theorem.

October 2026 claimed classification

Alberto Cagnetta’s preprint claims a classification covering the classical and logarithmic-potential cases and extending to general $p>1$ under Neumann or periodic conditions. If correct, it settles the proposed uniqueness picture and gives sharp parameter ranges, but the claim is unrefereed and independently unverified.

Current status (as of October 2026): The general classification is claimed in Cagnetta’s preprint but remains unverified, so the conjecture is not mathematically settled.

Boscaggin–Feltrin–Zanolin uniqueness conjecture

Primary source

Additional references

Progress summary

Solutions 0