gradient term

The objective of this paper is to investigate the elliptic singular Laplacian equation $\Delta u -|\nabla\,u|^{q}+u^{p}-u^{-\delta}=0$ in $\mathbb{R}^{N}$, where $N\geq1$, $1<q<p$ and $\delta>2$.  Our main contributions consist of establishing the existence of an entire strictly positive solution and analyzing certain properties of its asymptotic behavior, particularly when it exhibits monotonicity.