  We discuss recent advances in the theory of quasilinear equations of the type
$ -\Delta_{p} u = \sigma u^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case
$0<q< p-1$, where $\sigma$ is a nonnegative measurable function, or measure,
for the $p$-Laplacian $\Delta_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$, as
well as more general quasilinear, fractional Laplacian, and Hessian operators.
  Within this context, we obtain some new results, in particular, necessary and
sufficient conditions for the existence of solutions $u \in
\text{BMO}(\mathbb{R}^n)$, $u \in L^r_{{\rm loc}}(\mathbb{R}^n)$, etc., and
prove an enhanced version of Wolff's inequality for intrinsic nonlinear
potentials associated with such problems.
