  This article uses the iterative schemes and perturbation methods to
completely solve the Han-Li conjecture, i.e. the general boundary Yamabe
problem with prescribed constant scalar curvature and constant mean curvature
on compact manifolds $ (M, \partial M, g) $ with non-empty smooth boundary, $
\dim M \geqslant 3 $. It is equivalent to show the existence of a real,
positive, smooth solution of $ -\frac{4(n -1)}{n - 2} \Delta_{g} u + R_{g} u =
\lambda u^{\frac{n+2}{n - 2}} $ in $ M $, $ \frac{\partial u}{\partial \nu} +
\frac{n-2}{2} h_{g} u = \frac{n-2}{2} \zeta u^{\frac{n}{n - 2}} $ on $ \partial
M $ with some constants $ \lambda, \zeta \in \mathbb{R} $. This boundary Yamabe
problem is solved in cases according to the sign of the first eigenvalue $
\eta_{1} $ of the conformal Laplacian with homogeneous Robin condition. The
signs of scalar curvature $ R_{g} $ and mean curvature $ h_{g} $ play an
important role in this existence result. In contrast to the classical method,
the Weyl tensor and classification of boundary points play no role in the
proof. In addition, the method is not dimensional specific. The key steps
include a new version of monotone iteration scheme for nonlinear elliptic PDEs
and nonlinear boundary conditions, an introduction of the perturbed conformal
Laplacian, a delicate gluing skill to construct a smooth super-solution and
existence of solution of local Yamabe-type equations.
