Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms

TL;DR

This paper establishes bilateral pointwise estimates and existence criteria for positive solutions to a class of quasilinear elliptic equations with sub-natural growth terms, using Wolff potentials to separate the influences of source functions.

Contribution

It introduces a novel approach to separate the effects of source functions in bilateral estimates for solutions, extending previous results to more general operators and measures.

Findings

Derived bilateral pointwise estimates for solutions.

Separated contributions of source functions in estimates.

Extended results to general local and nonlocal operators.

Abstract

We study quasilinear elliptic equations of the type $-\Delta_{p} u = \sigma u^{q} + \mu \; \; \text{in} \;\; \bf{R}^n$ in the case $0<q< p-1$, where $\mu$ and $\sigma$ are nonnegative measurable functions, or locally finite measures, and $\Delta_{p}u= \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian. Similar equations with more general local and nonlocal operators in place of $\Delta_{p}$ are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions $u$: $$ u(x) \approx (\bf{W}_p \sigma(x))^{\frac{p-q}{p-q-1}} + \bf{K}_{p,q} \sigma(x) + \bf{W}_p \mu (x), \quad x \in \bf{R}^n,$$ where $\bf{W}_p$ and $\bf{K}_{p, q}$ are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on $p$, $q$ and $n$. The contributions of $\mu$ and $\sigma$ in these pointwise estimates are totally separated, which is a new phenomenon even when $p=2$. In the homogeneous case $\mu=0$, such estimates were obtained earlier by a different method only for minimal positive solutions.