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 \begin{equation} \label{main equation of the thesis} \left\{ \begin{array}{lr} -\Delta u = (1+\kappa_1(x)) | \nabla v |^p & \text{in}~~ B_1 \backslash \{0\},\\ -\Delta v = (1+\kappa_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 \geq 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.