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We discuss the $U(1)$ gauged Q-balls with $N$-power potential to examinetheir properties analytically. More numerical descriptions and some analyticalconsideration have been contributed to the models governed by four-powerpotential. We also demonstrate strictly some new limitations that the stable$U(1)$ gauged Q-balls should accept instead of estimating those with only somespecific values of model variables numerically. Having derived the explicitexpressions of radius, the Noether charge and energy of the gauged Q-balls, wefind that these models under the potential of matter field with general powerand the boundary conditions will exist instead of dispersing and decaying. TheNoether charge of the large gauged Q-balls must be limited. The mass parameterof the model can not be tiny.

The further analytical discussions on the $U(1)$ gauged Q-balls with $N$-power potential