On the existence of nonnegative solutions to semilinear differential inequality on Riemannian manifolds

(a) In this paper, we present a sufficient condition for the existence of solutions to semilinear elliptic inequality on a large class of manifolds. That is, if there exist constants r(0) and C-0 satisfying integral(+infinity)(2p-1)(r0)( t)/V-x(t)(p-1)dt <= C-0, for all x is an element of M, then there exists a positive solution to Delta u + u(p) <= 0, where V-x0 (r) is the volume of geodesic ball of radius r centered at x(0). (b). In Grigor'yan and Sun (2014), Grigor'yan and Sun proved that if V-x0 (r) <= r(alpha 1)(ln r)(alpha 2) then the only nonnegative weak solution of Delta u + u(p) <= 0 on a complete Riemannian manifold is identically 0, here V-x0 (r) is the volume of geodesic ball of radius r centered at x(0), and alpha(1) = 2p/p - 1, alpha(2) = 1/p - 1; moreover, parameters alpha(1) and alpha(2) are sharp that if alpha(2) > 1/p - 1 then there exists a manifold admiting a positive weak solution. In this paper, we present a sufficient condition for the existence of solutions on a large class of manifolds. That is, if there exist positive constants r(0) and C-0 satisfying integral(+infinity)(2p-1)(r0)( t)/V-x(t)(p - 1)dt <= C-0, for all x is an element of M, then there exists a positive solution to Delta u + u(p) <= 0. (C) 2019 Elsevier Ltd. All rights reserved.