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 \[ -Δ_{p} u = σu^{q} + μ\quad \text{in} \ \mathbb{R}^{n}, \] 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.

Published online in Nonlinear Analysis