  A second order ordinary differential equation with a superlinear term
$g(x,u)$ under radiation boundary conditions is studied. Using a shooting
argument, all the results obtained in a previous work for a Painlev\'e II
equation are extended. It is proved that the uniqueness or multiplicity of
solutions depend on the interaction between the mapping $\frac {\partial
g}{\partial u}(\cdot,0)$ and the first eigenvalue of the associated linear
operator. Furthermore, two open problems regarding, on the one hand, the
existence of sign-changing solutions and, on the other hand, exact multiplicity
are solved.
