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 {Delta u + u(tau)v(m) <= 0, Delta v + v(eta)u(n) <= 0, on a complete connected non-compact Riemannian manifold, where tau, eta >= 0, m, n > 0 are given parameters satisfying tau + m = eta + n = sigma > 1. We prove that, for some reference point x(0) if mu(B(x(0) , r)) <= Cr2 sigma/sigma-1 (ln r)(1/sigma-1) , holds for all large enough r. Then (1) admits only trivial solution. Here B(x(0), r) is a geodesic ball. We also show the sharpness of the volume growth condition (2). (C) 2017 Elsevier Inc. All rights reserved.