How does gravity save or kill Q-balls?

We explore stability of gravitating Q-balls with potential $V_4(\phi)={m^2\over2}\phi^2-\lambda\phi^4+\frac{\phi^6}{M^2}$ via catastrophe theory, as an extension of our previous work on Q-balls with potential $V_3(\phi)={m^2\over2}\phi^2-\mu\phi^3+\lambda\phi^4$. In flat spacetime Q-balls with $V_4$ in the thick-wall limit are unstable and there is a minimum charge $Q_{{\rm min}}$, where Q-balls with $Q<Q_{{\rm min}}$ are nonexistent. If we take self-gravity into account, on the other hand, there exist stable Q-balls with arbitrarily small charge, no matter how weak gravity is. That is, gravity saves Q-balls with small charge. We also show how stability of Q-balls changes as gravity becomes strong.