A comparison principle and applications to asymptotically $p$-linear boundary value problems

Abstract

Consider the problems \begin{equation*} \left\{ \begin{array}{@{}ll@{}} -\Delta_{p}u=f\ \text{in}\ \Omega{,} & u=0\ \text{on}\ \partial \Omega,\\ -\Delta_{p}v=g\ \text{in}\ \Omega{,} & v=0\ \text{on}\ \partial \Omega, \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega$, $\Delta_{p}z=\mathrm{div}(\lvert\nabla z\rvert^{p-2}\nabla z)$, $p>1$. We prove a strong comparison principle that allows $f-g$ to change sign. An application to singular asymptotically $p$-linear boundary problems is given.