Quasilinear elliptic equations with sub-natural growth terms and nonlinear potential theory

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.