  We study quasilinear elliptic equations of the type $-\Delta_{p} u = \sigma
u^{q} + \mu \; \; \text{in} \;\; \bf{R}^n$ in the case $0<q< p-1$, where $\mu$
and $\sigma$ are nonnegative measurable functions, or locally finite measures,
and $\Delta_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian.
Similar equations with more general local and nonlocal operators in place of
$\Delta_{p}$ are treated as well.
  We obtain existence criteria and global bilateral pointwise estimates for all
positive solutions $u$: $$ u(x) \approx (\bf{W}_p
\sigma(x))^{\frac{p-q}{p-q-1}} + \bf{K}_{p,q} \sigma(x) + \bf{W}_p \mu (x),
\quad x \in \bf{R}^n,$$ where $\bf{W}_p$ and $\bf{K}_{p, q}$ are, respectively,
the Wolff potential and the intrinsic Wolff potential, with the constants of
equivalence depending only on $p$, $q$ and $n$.
  The contributions of $\mu$ and $\sigma$ in these pointwise estimates are
totally separated, which is a new phenomenon even when $p=2$. In the
homogeneous case $\mu=0$, such estimates were obtained earlier by a different
method only for minimal positive solutions.
