Liouville theorem for the inequality Δₘ u+f(u)≤ 0 on Riemannian manifolds

In this paper, we study the quasilinear inequality Δₘ u+f(u)≤ 0 on a complete Riemannian manifold, where {align*} m>1,α>m-1 and f(t)> 0,αf(t)-tf'(t)≥ 0, ∀ t>0. {align*} If for some point x₀ and large enough r, {align*} vol B_r(x_0)≤ C r^p ln^q r, {align*} where p=mα/α-(m-1),q=m-1/α-(m-1) and Bᵣ(x₀) is a geodesic ball of radius r centered at x₀, then the inequality possesses no positive weak solution. This generalizes the result in {AS,Sun}.