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 $u \in \mathrm{BMO}(\mathbb R^n), u \in L^r_{{\mathrm loc}}(\mathbb R^n)$, etc., and prove an enhanced version of Wolff's inequality for intrinsic nonlinear potentials associated with such problems.