Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms

Abstract

We study the existence of positive solutions to quasilinear elliptic equations of the type \[ -\Delta_{p} u = \sigma u^{q} + \mu \quad \text{in} \ \mathbb{R}^{n}, \] in the sub-natural growth case $0 < q < p - 1$, where $\Delta_{p}u = \nabla \cdot ( |\nabla u|^{p - 2} \nabla u )$ is the $p$-Laplacian with $1 < p < n$, and $\sigma$ and $\mu$ are nonnegative Radon measures on $\mathbb{R}^{n}$. We construct minimal generalized solutions under certain generalized energy conditions on $\sigma$ and $\mu$. To prove this, we give new estimates for interaction between measures. We also construct solutions to equations with several sub-natural growth terms using the same methods.