Positive solutions to Δu-Vu+Wup=0 and its parabolic counterpart in noncompact manifolds

We consider the equation Deltau-V(x)u+W(x)u(p)=0 and its parabolic counterpart in noncompact manifolds. Under some natural conditions on the positive functions V and W, which may only have 'slow' or no decay near infinity, we establish existence of positive solutions in both the critical and the subcritical case. This leads to the solutions, in the difficult positive curvature case, of many scalar curvature equation in noncompact manifolds. The result is new even in the Euclidean space. In the subcritical, parabolic case, we also prove the convergence of some global solutions to nontrivial stationary solutions.