Absence of nonnegative solutions to the system of differential inequalities on manifolds

We mainly investigate the nonexistence of non-negative solution to the system of differential inequalities(1){Δu+uτvm≤0,Δv+vηun≤0, on a complete connected non-compact Riemannian manifold, where τ,η≥0, m,n>0 are given parameters satisfying τ+m=η+n=σ>1. We prove that, for some reference point x0 if(2)μ(B(x0,r))≤Cr2σσ−1(ln⁡r)1σ−1, holds for all large enough r. Then (1) admits only trivial solution. Here B(x0,r) is a geodesic ball. We also show the sharpness of the volume growth condition (2).