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In this article we study a class of prescribed curvature problems on completenoncompact Riemannian manifolds. To be precise, we derive local $C^0$-estimateunder an asymptotic condition which is in effect optimal, and prove theexistence of complete conformal metrics with prescribed curvature functions. Akey ingredient of our strategy is Aviles-McOwen's result or its fully nonlinearversion on the existence of complete conformal metrics with prescribedcurvature functions on manifolds with boundary.

Local $C^0$-estimate and existence theorems for some prescribed curvature problems on complete noncompact Riemannian manifolds