  The existence of self-similar solutions with fat tails for Smoluchowski's
coagulation equation has so far only been established for the solvable and the
diagonal kernel. In this paper we prove the existence of such self-similar
solutions for continuous kernels $K$ that are homogeneous of degree $\gamma \in
[0,1)$ and satisfy $K(x,y) \leq C (x^{\gamma} + y^{\gamma})$. More precisely,
for any $\rho \in (\gamma,1)$ we establish the existence of a continuous weak
self-similar profile with decay $x^{-(1{+}\rho)}$ as $x \to \infty$.
