  For $N\geq 3$, by the seminal paper of Brezis and V\'eron (Arch. Rational
Mech. Anal. 75(1):1--6, 1980/81), no positive solutions of $-\Delta u+u^q=0$ in
$\mathbb R^N\setminus \{0\}$ exist if $q\geq N/(N-2)$; for $1<q<N/(N-2)$ the
existence and profiles near zero of all positive $C^1(\mathbb R^N\setminus
\{0\})$ solutions are given by Friedman and V\'eron (Arch. Rational Mech. Anal.
96(4):359--387, 1986).
  In this paper, for every $q>1$ and $\theta\in \mathbb R$, we prove that the
nonlinear elliptic problem (*) $-\Delta u-\lambda
\,|x|^{-2}\,u+|x|^{\theta}u^q=0$ in $\mathbb R^N\setminus \{0\}$ with $u>0$ has
a $C^1(\mathbb R^N\setminus \{0\})$ solution if and only if
$\lambda>\lambda^*$, where $\lambda^*=\Theta(N-2-\Theta) $ with
$\Theta=(\theta+2)/(q-1)$. We show that (a) if $\lambda>(N-2)^2/4$, then
$U_0(x)=(\lambda-\lambda^*)^{1/(q-1)}|x|^{-\Theta}$ is the only solution of (*)
and (b) if $\lambda^*<\lambda\leq (N-2)^2/4$, then all solutions of (*) are
radially symmetric and their total set is $U_0\cup \{U_{\gamma,q,\lambda}:\
\gamma\in (0,\infty) \}$. We give the precise behavior of $
U_{\gamma,q,\lambda}$ near zero and at infinity, distinguishing between
$1<q<q_{N,\theta}$ and $q>\max\{q_{N,\theta},1\}$, where
$q_{N,\theta}=(N+2\theta+2)/(N-2)$.
  In addition, for $\theta\leq -2$ we settle the structure of the set of all
positive solutions of (*) in $\Omega\setminus \{0\}$, subject to
$u|_{\partial\Omega}=0$, where $\Omega$ is a smooth bounded domain containing
zero, complementing the works of C\^{\i}rstea (Mem. Amer. Math. Soc. 227, 2014)
and Wei--Du (J. Differential Equations 262(7):3864--3886, 2017).
