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

TL;DR

This paper investigates conditions for the existence of positive solutions to weighted quasilinear elliptic equations with sub-natural growth terms in bounded domains, focusing on measure data and properties of p-superharmonic functions.

Contribution

It provides new criteria for existence of solutions to weighted p-Laplacian equations with measure data in the sub-natural growth regime.

Findings

Established existence criteria for solutions with measure data.

Analyzed properties of p-superharmonic functions relevant to solvability.

Extended understanding of Dirichlet problems with infinite measure data.

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.