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

We study quasilinear elliptic equations of the type $-\Delta_{p} u = \sigma u^{q} + \mu ; ; $\text{in}$ ;; \bf{R}^n$inthecase$0<q< p-1$,where$\mu$and$\sigma$ are nonnegative measurable functions, or locally finite measures, and ${\mathrm{\Delta }}_{p}u=\text{div}(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)$ is the $p$-Laplacian. Similar equations with more general local and nonlocal operators in place of ${\mathrm{\Delta }}_p$ are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions $u$: $$u(x)\approx ({\mathbf{W}}_{\mathbf{p}}\mathbf{\sigma }(\mathbf{x}){)}^{\frac{\mathbf{p}-\mathbf{q}}{\mathbf{p}-\mathbf{q}-\mathbf{1}}}+{\mathbf{K}}_{\mathbf{p},\mathbf{q}}\mathbf{\sigma }(\mathbf{x})+{\mathbf{W}}_{\mathbf{p}}\mathbf{\mu }(\mathbf{x}),\mathbf{x}\in {\mathbf{R}}^{\mathbf{n}},$$ where ${\mathbf{W}}_{\mathbf{p}}$ and ${\mathbf{K}}_{\mathbf{p},\mathbf{q}}$ 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.