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

We prove that any nonnegative viscosity solution of the inequality ( − Δ p ) s u ≥ u t | ∇ u | m in R N , N ≥ 2 , \begin{equation*} (-\Delta _p)^s u \geq u^{t} |\nabla u|^{m}\quad \text { in }\; \mathbb {R}^N,\; N\geq 2, \end{equation*} must be constant. This result holds for parameters p ∈ ( 1 , ∞ ) , s ∈ ( 0 , 1 ) p\in (1, \infty ), s\in (0, 1) , t , m ≥ 0 t, m\geq 0 , satisfying t ( N − s p ) + m ( N − ( s p − p + 1 ) ) > N ( p − 1 ) , \begin{equation*} t (N-sp) + m(N-(sp-p+1)) > N(p-1), \end{equation*} with the additional condition that either m ≤ p − 1