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

TL;DR

This paper establishes Liouville-type theorems for nonnegative solutions of fractional p-Laplacian inequalities with gradient nonlinearities, showing solutions must be constant under certain parameter conditions.

Contribution

It proves new Liouville results for fractional p-Laplacian inequalities involving gradient nonlinearities, extending previous understanding to a broader parameter range.

Findings

Nonnegative solutions are constant under specified conditions.

Conditions relate parameters p, s, t, m, and dimension N.

Results apply to a wide class of fractional p-Laplacian inequalities.

Abstract

We prove that any nonnegative viscosity solution of the inequality $$(-\Delta_p)^s u(x) \geq u^{t} |\nabla u|^{m}\quad \text{ in }\; \mathbb{R}^N,\; N\geq 2,$$ 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 $p-1\geq sp$.