How does gravity save or kill Q-balls?

We explore stability of gravitating $Q$-balls with potential ${V}_{4}(\ensuremath{\phi})=\frac{{m}^{2}}{2}{\ensuremath{\phi}}^{2}\ensuremath{-}\ensuremath{\lambda}{\ensuremath{\phi}}^{4}+\frac{{\ensuremath{\phi}}^{6}}{{M}^{2}}$ via catastrophe theory, as an extension of our previous work on $Q$-balls with potential ${V}_{3}(\ensuremath{\phi})=\frac{{m}^{2}}{2}{\ensuremath{\phi}}^{2}\ensuremath{-}\ensuremath{\mu}{\ensuremath{\phi}}^{3}+\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$. In flat spacetime $Q$-balls with ${V}_{4}$ in the thick-wall limit are unstable and there is a minimum charge ${Q}_{\mathrm{min}}$, where $Q$-balls with $Ql{Q}_{\mathrm{min}}$ 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.