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

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