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

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.