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Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary$\partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and$\partial M$, respectively, it is natural to ask whether there exist metricsconformal to $g$ so that under these new metrics, $K$ is the scalar curvatureand $c$ is the boundary mean curvature. All such metrics can be described by aprescribing curvature equation with a boundary condition. With suitableassumptions on $K$,$c$ and $(M,g)$ we show that all the solutions of theequation can only blow up at finite points over each compact subset of $\barM$, some of them may appear on $\partial M$. We describe the asymptoticbehavior of the blowup solutions around each blowup point and derive an energyestimate as a consequence.

Prescribing curvatures on three dimensional Riemannian manifolds with boundaries