Symmetry and monotonicity property of a solution of \((p,q)\) Laplacian equation with singular terms (Q6203816)

Summary: 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 \[ \begin{cases} -\mathrm{div} (|\nabla u|^{p-2} \nabla u+a(x)|\nabla u|^{q-2}\nabla u)=\frac{g(x)}{u^{\delta}}+h(x)f(u) & \text{in }B_R (x_0), \\ u=0 & \text{on }\partial B_R (x_0). \end{cases} \] 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 \(f\equiv 1\).