  We consider a branching random walk with an absorbing barrier, where the step
of the associated one-dimensional random walk is in the domain of attraction of
an $\alpha$-stable law with $1<\alpha<2$. We shall prove that there is a
barrier $an^{\frac{1}{1+\alpha}}$ and a critical value $a_\alpha$ such that if
$a<a_\alpha$, then the process dies; if $a>a_\alpha$, then the process
survives. The results generalize previous results in literature for the case
$\alpha=2$.
