  We introduce an iterative scheme to prove the Yamabe problem $ - a\Delta_{g}
u + S u = \lambda u^{p-1} $, firstly on open domain $ (\Omega, g) $ with
Dirichlet boundary conditions, and then on closed manifolds $ (M, g) $ by local
argument. It is a new proof, which solves the Yamabe problem for $ n \geqslant
3 $ in a uniform argument, beyonds the traditional analysis with respect to the
minimization of functionals.
