  We study radial solutions of the semilinear elliptic equation $\Delta
u+f(u)=0$ under rather general growth conditions on $f$. We construct a radial
singular solution and study the intersection number between the singular
solution and a regular solution. An application to bifurcation problems of
elliptic Dirichlet problems is given. To this end, we derive a certain limit
equation from the original equation at infinity, using a generalized similarity
transformation. Through a generalized Cole-Hopf transformation, all the limit
equations can be reduced into two typical cases, i.e., $\Delta u+u^p=0$ and
$\Delta u+e^u=0$.
