Strong comparison principle for subquadratic fractional p-Laplacian equations in the general parameter regime

Establish a strong comparison principle for weak solutions of subquadratic fractional p-Laplacian equations throughout the parameter range 1<p<2, beyond the restricted regime s(0,1/2) and 1/(1-s)<p<2, under suitably mild regularity assumptions.

Background

The paper identifies the strong comparison principle for weak solutions of fractional p-Laplacian equations as unresolved in the subquadratic regime. Its own theorem proves the principle only for continuous weak solutions with s(0,1/2) and 1/(1-s)<p<2. Thus, the broader problem remains open for parameter values and solution classes not covered by that theorem.

The principal difficulty is the singularity of the fractional p-Laplacian when 1<p<2, which complicates the integral estimates needed to compare a solution with its reflection. Earlier results treated only special equations or imposed conditions that prevent direct application to reflected solutions.

References

Despite the above developments, the strong comparison principle of weak solutions to main in the subquadratic regime $1<p<2$ remained widely open.

main:

$\begin{aligned} \begin{cases} (-\Delta)_p^s u= f(u)\quad&\mbox{in}\; \Omega\\ u=0&\mbox{in}\; \mathbb{R}^N\backslash \Omega,\\ \end{cases} \end{aligned} $

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