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

We study the existence of positive solutions to quasilinear elliptic equations of the type $$-{\mathrm{\Delta }}_{p}u=\sigma u^q+\mu \text{in }{\mathbb{R}}^n,$$ in the sub-natural growth case $0<q<p-1$, where ${\mathrm{\Delta }}_{p}u=\mathrm{\nabla }\cdot (\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)$ 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.