  We show the existence of self-similar solutions with fat tails for
Smoluchowski's coagulation equation for homogeneous kernels satisfying $C_1
\left(x^{-a}y^{b}+x^{b}y^{-a}\right)\leq K\left(x,y\right)\leq
C_2\left(x^{-a}y^{b}+x^{b}y^{-a}\right)$ with $a>0$ and $b<1$. This covers
especially the case of Smoluchowski's classical kernel $K(x,y)=(x^{1/3} +
y^{1/3})(x^{-1/3} + y^{-1/3})$.
  For the proof of existence we first consider some regularized kernel
$K_{\epsilon}$ for which we construct a sequence of solutions $h_{\epsilon}$.
In a second step we pass to the limit $\epsilon\to 0$ to obtain a solution for
the original kernel $K$. The main difficulty is to establish a uniform lower
bound on $h_{\epsilon}$. The basic idea for this is to consider the
time-dependent problem and choosing a special test function that solves the
dual problem.
