This paper examines the behavior of a positive solution $u\in C^{ 1,\alpha}(\Bar{ \Omega})$ 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^\delta}+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 $0<\delta<1$, $1<p\leq q<\infty$, 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 $1<p\leq q\leq 2$ and $f\equiv1$.
Categories: math.AP