On the existence of nonnegative solutions to semilinear differential inequality on Riemannian manifolds. (June 2019) - Xu, Fanheng
Abstract: (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 ∫ r 0 + ∞ t 2 p − 1 V x ( t ) p − 1 d t ⩽ C 0, ∀ x ∈ M, then there exists a positive solution to Δ u + u p ⩽ 0, where V x 0 ( 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 x 0 ( r ) ⩽ r α 1 ( ln r ) α 2 then the only nonnegative weak solution of Δ u + u p ⩽ 0 on a complete Riemannian manifold is identically 0, here V x 0 ( r ) is the volume of geodesic ball of radius r centered at x 0, and α 1 = 2 p p − 1, α 2 = 1 p − 1 ; moreover, parameters α 1 and α 2 are sharp that if α 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 ∫ r 0 + ∞ t 2 p − 1 V x ( t ) p − 1 d t ⩽ C 0, ∀ x ∈ M, then there exists a positive solution to Δ u + u p ⩽ 0 .