Symmetry and monotonicity property of a solution of $(p,q)$ Laplacian equation with singular terms

Abstract

This paper examines the behavior of a positive solution $u\in C^{1,\alpha }(\overline{B_R(x_0)})$ of the $(p,q)$ Laplace equation with a singular term and zero Dirichlet boundary condition. Specifically, we consider the equation

We assume that $0<\delta <1$, $1<p\le q<\infty$, and $f$ is a $C^1(\mathbb{R})$ nondecreasing function. Our analysis uses the moving plane method to investigate the symmetry and monotonicity properties of $u$. Additionally, we establish a strong comparison principle for solutions of the $(p,q)$ Laplace equation with radial symmetry under the assumptions that $1<p\le q\le 2$ and $f\equiv 1$.