Positive singular solutions of a certain elliptic PDE

Abstract

In this paper, we investigate the existence of positive singular solutions for a system of partial differential equations on a bounded domain(1)$\{\begin{array}{cc}-\Delta u=(1+{\kappa }_1(x\left)\right)|\nabla v{|}^p& \text{in}{B}_1﹨\left\{0\right\},\\ -\Delta v=(1+{\kappa }_2(x\left)\right)|\nabla u{|}^p& \text{in}{B}_1﹨\left\{0\right\},\\ u=v=0& \text{on}\partial B_1.\end{array}$ We investigate the existence of positive singular solutions within $B_1$, the unit ball centered at the origin in $R^N$, 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 ${\kappa }_1\left(0\right)={\kappa }_2\left(0\right)=0$. Our system is an extension of the PDE studied by Aghajani et al. [1] under similar assumptions.