How does gravity save or kill Q-balls?

We explore stability of gravitating Q-balls with potential V-4(phi) = m(2)/2 phi(2) - lambda phi(4) + phi(6)/M-2 via catastrophe theory, as an extension of our previous work on Q-balls with potential V-3(phi) m(2)/2 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(min), where Q-balls with Q < Q(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.