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

TL;DR

This paper establishes a strong comparison principle for certain singular p-Laplace equations with radially decreasing solutions, demonstrating its validity for 1<p<2 and providing applications such as a three-solution theorem, while also showing its failure for p>2.

Contribution

The paper proves a strong comparison principle for singular p-Laplace equations with specific conditions and introduces applications, including a three-solution theorem, highlighting differences for p>2.

Findings

Strong comparison principle holds for 1<p<2 in singular p-Laplace equations.

Counterexample shows violation of the principle for p>2.

Application includes a three-solution theorem for p-Laplace equations.

Abstract

In this paper we prove a strong comparison principle for radially decreasing solutions $u,v\in C_{0}^{1,\alpha}(\Bar{B_R})$ 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 $0 \leq f \leq g$ 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.