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

Abstract

We study quasilinear elliptic equations of the type $-{\Delta }_{p}u=\sigma u^q+\mu \text{in}{\mathbf{R}}^{\mathbf{n}}$ 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 $u$: $$u(x)\approx ({\mathbf{W}}_{\mathbf{p}}\sigma (\mathbf{x}){)}^{\frac{\mathbf{p}-\mathbf{q}}{\mathbf{p}-\mathbf{q}-1}}+{\mathbf{K}}_{\mathbf{p},\mathbf{q}}\sigma (\mathbf{x})+{\mathbf{W}}_{\mathbf{p}}\mu (\mathbf{x}),\mathbf{x}\in {\mathbf{R}}^{\mathbf{n}},$$ 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.

arXiv comment18 pages