Positive singular solutions of a certain elliptic PDE

In this paper, we investigate the existence of positive singular solutions for a system of partial differential equations on a bounded domain \begin{equation} \label{main equation of the thesis} \left{ \begin{array}{lr} -Δu = (1+κ_1(x)) | \nabla v |^p & $\text{in}$~~ B_1 \backslash {0},\ -Δv = (1+κ_2(x)) | \nabla u |^p & $\text{in}$~~ B_1 \backslash {0},\ u = v = 0 & $\text{on}$~~ \partial B_1. \end{array} \right. \end{equation} We investigate the existence of positive singular solutions within $B_1$, the unit ball centered at the origin in ${\mathbb{R}}^N$, under the conditions $N\ge 3$ and $\frac{N}{N-1}<p<2$. Additionally, we assume that ${\kappa }_1$ and ${\kappa }_2$ are non-negative, continuous functions satisfying ${\kappa }_1(0)={\kappa }_2(0)=0$ . Our system is an extension of the PDE equation studied by Aghajani et al. (2021) under similar assumptions.