Quasilinear elliptic equations with sub-natural growth terms in bounded domains

Abstract

We consider the existence of positive solutions to weighted quasilinear elliptic differential equations of the type \[ \begin{cases} - \Delta_{p, w} u = \sigma u^{q} & \text{in $\Omega$}, \\ u = 0 & \text{on $\partial \Omega$} \end{cases} \] in the sub-natural growth case $0 < q < p - 1$, where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$, $\Delta_{p, w}$ is a weighted $p$-Laplacian, and $\sigma$ is a nonnegative (locally finite) Radon measure on $\Omega$. We give criteria for the existence problem. For the proof, we investigate various properties of $p$-superharmonic functions, especially the solvability of Dirichlet problems with infinite measure data.