For the generalized Thomas--Fermi differential equation \[ (|x'|^{lpha-1}x')'=q(t)|x|^{\beta-1}x, \] it is proved that if $1 \leq \alpha<\beta$ and $q(t)$ is a regularly varying function of index $\mu$ with $\mu>-\alpha-1$, then all positive solutions that tend to zero as $t\to\infty$ are regularly varying functions of one and the same negative index $\rho$ and their asymptotic behavior at infinity is governed by the unique definite decay law. Further, an attempt is made to generalize this result to more general quasilinear differential equations of the form \[ (p(t)|x'|^{lpha-1}x')'=q(t)|x|^{\beta-1}x. \]