Liouville results for supersolutions of fractional $p$-Laplacian equations with gradient nonlinearities

TL;DR

This work establishes a Liouville-type rigidity result for nonlocal, nonlinear inequalities involving the fractional $p$-Laplacian with gradient nonlinearities. The authors develop a self-contained iterative bootstrap that refines lower bounds for positive supersolutions, leveraging barrier constructions and a viscosity-solution framework to handle the gradient term. Under the subcritical balance $t(N-sp)+m(N-(sp-p+1))<N(p-1)$ and appropriate bounds on $m$ relative to $sp$ and $p$, they show every nonnegative solution on $\mathbb{R}^N$ must be constant, extending previously known results for the $m=0$ case and addressing the nonlocal gradient-nonlinearity gap. The methodology provides a robust approach for nonlocal Liouville problems and informs regularity/blow-up analyses in fractional quasilinear settings.

Abstract

We prove that any nonnegative viscosity solution of the inequality $$(-Δ_p)^s u(x) \geq u^{t} |\nabla u|^{m}\quad \text{ in }\; \mathbb{R}^N,\; N\geq 2,$$ must be constant. This result holds for parameters $p\in (1, \infty), s\in (0, 1)$, $t, m\geq 0$, satisfying $$t (N-sp) + m(N-(sp-p+1)) < N(p-1),$$ with the additional condition that either $m\leq p-1$ if $p-1<sp$, or $m<sp$ if $p-1\geq sp$.

Theorems & Definitions (18)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Definition 2.1
• Remark 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3