Solving Yamabe Problem by An Iterative Method

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.