We introduce an iterative scheme to prove the Yamabe problem $ - aDelta_{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.
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