What happens to Q-balls if $Q$ is so large?

In the system of a gravitating Q-ball, there is a maximum charge $Q_{{rm max}}$ inevitably, while in flat spacetime there is no upper bound on $Q$ in typical models such as the Affleck-Dine model. Theoretically the charge $Q$ is a free parameter, and phenomenologically it could increase by charge accumulation. We address a question of what happens to Q-balls if $Q$ is close to $Q_{{rm max}}$. First, without specifying a model, we show analytically that inflation cannot take place in the core of a Q-ball, contrary to the claim of previous work. Next, for the Affleck-Dine model, we analyze perturbation of equilibrium solutions with $Qapprox Q_{{rm max}}$ by numerical analysis of dynamical field equations. We find that the extremal solution with $Q=Q_{{rm max}}$ and unstable solutions around it are critical solutions, which means the threshold of black-hole formation.