Radial symmetry of minimizers for some translation and rotation invariant functionals (Q1906168)

The author has studied the radial symmetry of the minimizers of the functional \[ \textstyle{{1\over 2}} \int_{\mathbb{R}^N} |\text{grad } u(x)|^2 dx+ \int_{\mathbb{R}^N} F(u(x)) dx \] subject to \(\int_{\mathbb{R}^N} G(u(x))dx= 1\). This question has been studied without positivity assumption. The proof works for more general functionals.