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Q-balls are non-topological solitons in a large family of field theories. Wefocus on the existence of $U(1)$ gauged Q-balls for a field theory withsixth-order potential. The problem can be reduced to proving the existence ofcritical points for some indefinite functional. For this, we use a constrainedminimization approach to obtain the existence of critical points. Moreover, weestablish some qualitative properties of the Q-ball solution, such asmonotonicity, boundedness and asymptotic behavior.

Existence of $U(1)$ Gauged Q-balls for A Field Model with Sixth-order Potential