Symmetry and Monotonicity Property of a Solution of (p,q) Laplace Equation with Singular Term

This paper examines the behavior of a positive solution $u\in C^{1,\alpha }(\text{Bar}\mathrm{\Omega })$ of the $(p,q)$ Laplace equation with a singular term and zero Dirichlet boundary condition. Specifically, we consider the equation: \begin{equation*} -div(|\nabla u|^{p-2}\nabla u+ a(x) |\nabla u|^{q-2}\nabla u) &= \frac{g(x)}{u^δ}+h(x)f(u) , &$\text{in}$ \thinspace B_R(x_0), \quad u & =0 \ &$\text{on}$ \ \partial B_R(x_0). \end{equation*} We assume that $0<\delta <1$, $1<p\le q<\mathrm{\infty }$, 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 $f\equiv 1$.