  The main purpose of this paper is to capture the asymptotic behavior for
solutions to a class of nonlinear elliptic and parabolic equations with the
anisotropic weights consisting of two power-type weights of different
dimensions near the degenerate or singular point, especially covering the
weighted $p$-Laplace equations and weighted fast diffusion equations. As a
consequence, we also establish the local H\"{o}lder estimates for their
solutions in the presence of single power-type weights.
