Existence and uniqueness of solutions for a class of $p$-Laplace equations on a ball

TL;DR

This paper investigates conditions under which positive solutions to a class of p-Laplace equations on a ball exist and are unique, extending results to a more general setting when the dimension is one.

Contribution

It provides new criteria for existence and uniqueness of solutions for generalized p-Laplace equations, including a broader case for one-dimensional domains.

Findings

Established existence conditions for positive solutions.

Proved uniqueness under certain conditions.

Extended results to the one-dimensional case.

Abstract

For a class of equations generalizing the model case \[ \Delta _p u-a(r)u^{p-1}+b(r)u^q=0 \; \; \mbox{in $B$}, \; \; u=0 \; \; \mbox{on $\partial B$}, \] where $B$ is the unit ball in $R^n$, $n \geq 1$, $r=|x|$, $p,q>1$, and $\Delta _p$ denotes the $p$-Laplace operator, we give conditions for the existence and uniqueness of positive solution. In case $n=1$, we give a more general result.