Using the finite-size scaling, we have investigated the percolation phase transitions of evolving random networks under a generalized Achlioptas process (GAP). During this GAP, the edge with minimum product of two connecting cluster sizes is taken with a probability $p$ from two randomly chosen edges. This model becomes the ErdH os-Renyi network at $p=0.5$ and the random network under the Achlioptas process at $p=1$. Using both the fixed point of $s_2/s_1$ and the straight line of $ln s_1$, where $s_1$ and $s_2$ are the reduced sizes of the largest and the second largest cluster, we demonstrate that the phase transitions of this model are continuous for $0.5 le p le 1$. From the slopes of $ln s_1$ and $ln (s_2/s_1)$ at the critical point we get the critical exponents $beta$ and $ u$, which depend on $p$. Therefore the universality class of this model should be characterized by $p$ also.
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