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

## TL;DR

This paper advances the understanding of quasilinear elliptic equations with sub-natural growth terms, providing new existence criteria and improved inequalities for solutions involving nonlinear potential theory.

## Contribution

It introduces necessary and sufficient conditions for solutions in various function spaces and enhances Wolff's inequality for nonlinear potentials in this context.

## Key findings

- Established criteria for solution existence in BMO and local L^r spaces.
- Proved an improved version of Wolff's inequality for nonlinear potentials.
- Extended results to fractional Laplacian and Hessian operators.

## Abstract

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.