On the Phragm\'{e}n-Lindel\"{o}f and the superposition principles for the $p$-Laplacian

Abstract

We study sub and supersolutions for the $p$-Laplace type elliptic equation of the form $$-\Delta_p u-V|u|^{p-2}u=0\quad\text{in $\Omega$},$$ where $\Omega$ is a radially symmetric domain in ${\mathbb{R}}^N$ and $V(x)\ge 0$ is a continuous potential such that the solutions of the equation satisfy the comparison principle on bounded subdomains of $\Omega$. In this work we establish a superposition principle and then use it to develop a version of a Phragm\'{e}n-Lindel\"{o}f comparison principle in the case $p\ge 2$. Moreover, by applying this principle to the case of Hardy-type potentials we recover and improve a number of known lower and upper estimates for sub and supersolutions.