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

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