This paper is contributed to the inhomogeneous semilinear elliptic equation

\begin{equation} \Delta u+K(|x|)u^{p}+f(x)=0\quad\text{in }\mathbb{R}^n, \tag{$*$} \label{*} \end{equation}

where

$$ \Delta=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}} $$

is the Laplacian operator, $n\geq3$, $p>1$, $f(x)\geq0$ and $K(|x|)>0$ is a given locally Hölder continuous function in $\mathbb{R}^n\setminus\{0\}$. The existence, non-existence and decay properties of positive solutions for \eqref{*} are obtained under some assumptions on $f(x)$ and $K(|x|)$ satisfying the slow-decay condition, i.e. $K(|x|)\geq C|x|^{l}$ at infinity for some constants $C>0$ and $l>-2$. The decay properties of positive solutions for $(\ast)$ are also discussed for the critical decay case on $K(|x|)$ with $l=-2$.