It has been a classical question which manifolds admit Riemannian metrics with positive scalar curvature. A manifold is called aspherical if it has contractible universal cover. By works of Chodosh—Li, Gromov, and Chodosh—Li—Liokumovich, for n = 4, 5, the connected sum of a closed aspherical n-manifold with an arbitrary closed manifold does not admit a metric with positive scalar curvature. We prove for n = 3,4,5 that the connected sum of a closed aspherical n-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. In particular, a special case of our result answers a question of Gromov. This is joint work with Jianchun Chu and Jintian Zhu.