Strong comparison principle for a p-Laplace equation involving singularity and its applications

Abstract

In this paper we prove a strong comparison principle for radially decreasing solutions $u,v\in C_{0}^{1,α}(\Bar{B_R})$ of the singular equations $-{\Delta }_{p}u-\frac{1}{u^{\delta }}=f(x)$ and $-{\Delta }_{p}v-\frac{1}{v^{\delta }}=g(x)$ in $B_R$. Here we assume that $1<p<2,\delta \in (0,1)$ and $f,g$ are continuous, radial functions such that $0\le f\le g$ but $f\not\equiv g$ in $B_R.$ For the case $p>2$ a counterexample is provided where the strong comparison principle is violated. As an application of strong comparison principle, we prove a three solution theorem for p-Laplace equation and illustrate with an example.

9 pages, 0 figure