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 $-{\mathrm{\Delta }}_{p}u=\sigma u^q\text{in}{\mathbb{R}}^n,$ in the case $0<q<p-1$, where $\sigma$ is a nonnegative measurable function, or measure, for the $p$-Laplacian ${\mathrm{\Delta }}_{p}u=\text{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)$, 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.