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We employ three different methods to prove the following result on prescribedscalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a$n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assumethe conformal invariant $Y(M,\partial M)<0$. Given any negative smoothfunctions $f$ in $M$ and $h$ on $\partial M$, there exists a unique conformalmetric of $g_0$ such that its scalar curvature equals $f$ and mean curvaturecurvature equals $h$. The first two methods are sub-super-solution method andsubcritical approximation, and the third method is a geometric flow. In theflow approach, assume another conformal invariant $Q(M,\pa M)$ is a negativereal number, for some class of initial data, we prove the short time and longtime existences of the so-called prescribed scalar curvature plus meancurvature flows, as well as their asymptotic convergence. Via a family of suchflows together with some additional variational arguments, under the flowassumptions we prove existence and uniqueness of positive minimizers of theassociated energy functional and also the above result by analyzing asymptoticlimits of the flows and the relations among some conformal invariants.

Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant