In studies of superlinear parabolic equations
$$
u_t=\Delta u+u^p,\quad x\in \mathbb R^N,\ t>0,
$$
where $p>1$, backward self-similar solutions play an important
role. These are solutions of the form $ u(x,t) =
(T-t)^{-1/(p-1)}w(y)$, where
$y:=x/\sqrt{T-t}$, $T$ is a constant, and $w$ is a solution of the
equation
$\Delta w-y\cdot\nabla w/2 -w/(p-1)+w^p=0$. We consider (classical)
positive radial solutions $w$ of this equation. Denoting by $p_S$,
$p_{JL}$,
$p_L$ the Sobolev, Joseph-Lundgren, and Lepin exponents,
respectively,
we show that for $p\in (p_S,p_{JL})$ there are only countably many
solutions,
and for $p\in (p_{JL},p_L)$ there are only finitely many solutions.
This result answers two basic open questions regarding the
multiplicity
of the solutions.