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

Abstract

We discuss recent advances in the theory of quasilinear equations of the type $-{\Delta }_{p}u=\sigma u^q$ in ${\mathbb{R}}^n$, in the case $0<q<p-1$, where $\sigma$ is a nonnegative measurable function, or measure, for the $p$-Laplacian ${\Delta }_{p}u=\mathrm{div}(|\nabla u{|}^{p-2}\nabla u)$, 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 $u\in \mathrm{BMO}({\mathbb{R}}^n),u\in L_{\mathrm{l}oc}^r({\mathbb{R}}^n)$, etc., and prove an enhanced version of Wolff's inequality for intrinsic nonlinear potentials associated with such problems.