Liouville type theorem for quasi-linear elliptic inequality Δpu+uσ⩽0 on Riemannian manifolds

We studied the uniqueness of a nonnegative solution of the quasi-linear elliptic inequality(⁎)Δpu+uσ⩽0 on a connected geodesically complete Riemannian manifold X, where σ is a parameter and Δpu=div(|∇u|p−2∇u). We proved that if p>1, σ>p−1, and for some x0∈X,liminft→0+tσσ−p+1∫1∞μ(B(x0,r))r(p+1)σ−p+1σ−p+1+pt2(p−1)dr<∞, then inequality (⁎) has no nontrivial nonnegative weak solutions. We also studied the weighed case and the sharpness of the main result.