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

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