  In this paper we study a class of coagulation equations including a source
term that injects in the system clusters of size of order one. The coagulation
kernel is homogeneous, of homogeneity $\gamma < 1$, such that $K(x,y)$ is
approximately $x^{\gamma + \lambda} y^{-\lambda}$, when $x$ is larger than $y$.
We restrict the analysis to the case $\gamma + 2 \lambda \geq 1 $. In this
range of exponents, the transport of mass toward infinity is driven by
collisions between particles of different sizes. This is in contrast with the
case when $\gamma + 2 \lambda <1$. In that case, the transport of mass toward
infinity is due to the collision between particles of comparable sizes. In the
case $\gamma+2\lambda\geq 1$, the interaction between particles of different
sizes leads to an additional transport term in the coagulation equation that
approximates the solution of the original coagulation equation with injection
for large times. We prove the existence of a class of self-similar solutions
for suitable choices of $\gamma $ and $\lambda$ for this class of coagulation
equations with transport. We prove that for the complementary case such
self-similar solutions do not exist.
