  We introduce a double iterative scheme and local variational method to solve
the Yamabe-type equation $ - \frac{4(n - 1)}{n - 2}\Delta_{g} u + (S_{g} +
\beta ) u = \lambda u^{\frac{n + 2}{n - 2}} $ for some constant $ \beta
\leqslant 0 $, locally on Riemannian domain $ (\Omega, g) $ with trivial
Dirichlet condition and globally on closed manifolds $ (M, g) $; the dimensions
of $ \Omega $ and $ M $ are at least 3. In contrast to the traditional global
variational method, these Yamabe-type equations on closed manifolds are
analyzed by local analysis and monotone iteration scheme. In particular, we do
not need to use the Weyl tensor. In particular, the sign of the first
eigenvalue $ \eta_{1} $ of conformal Laplacian $ \Box_{g} u : = - \frac{4(n -
1)}{n - 2}\Delta_{g} u + S_{g} u $ plays the central role. When letting $ \beta
\rightarrow 0 $ from the left, we reproof the classical Yamabe problem as a
natural consequence of the Yamabe-type equations. The results are classified by
the sign of $ \eta_{1} $.
