  In this paper, we investigate the noncompact prescribed Chern scalar
curvature problem which reduces to solve a Kazdan-Warner type equation on
noncompact non-K\"{a}hler manifolds. By introducing an analytic condition on
noncompact manifolds, we establish related existence results. As its another
application, we further give a new proof of a classical multiplicity theorem of
W.M. Ni.

Differential Geometry;Analysis of PDEs