Strong comparison principle for the classical p-Laplacian in higher dimensions

Establish whether two ordered p-harmonic functions in a domain in [?] can satisfy only the alternatives of strict ordering throughout the domain or equality everywhere when p !=2 and N3; equivalently, determine whether uv or u>v in the domain follows from uv for p-harmonic functions.

Background

The paper discusses the strong comparison principle for ordered p-harmonic functions u and v satisfying p u=p v=0 and uv in a domain. In the plane, a general affirmative result is known, but the corresponding higher-dimensional case for p2 has not been resolved. This question concerns whether contact between two ordered p-harmonic functions forces global equality, or whether strict inequality must hold everywhere unless the functions coincide.

The issue is presented as a longstanding obstacle in the theory of quasilinear elliptic equations and provides context for the papers study of analogous comparison principles for the fractional p-Laplacian.

References

The only general affirmative answer was given by Manfredi in the plane $2$, and the situations for $p \ne 2$, $N\ge 3$ remain a famous longstanding open question.

— Comparison principles and symmetry for subquadratic fractional $p$-Laplacian equations  (2609.02626 - Ye et al., 2 Sep 2026) in Section 1, Introduction