  In studies of superlinear parabolic equations \begin{equation*}
  u_t=\Delta u+u^p,\quad x\in {\mathbb R}^N,\ t>0, \end{equation*} 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.
