Radial and nonradial minimizers for some radially symmetric functionals

$$ V(u) = {1over 2}int_{R^N} |{ m grad}, u(x)|^2, dx + int_{R^N}F(u(x)),dx $$ subject to $$ int_{R^N} G(u(x)), dx = lambda > 0,$$ where $u(x) = (u_1(x) , ldots, u_K(x))$ belongs to $H^1_K (R^N) = H^1 (R^N) imescdotsimes H^1(R^N)$ (K times) and $|{ m grad}, u(x)|^2$ means $ sum^K_{i=1}|{ m grad}, u_i (x)|^2$. We have shown that, under some technical assumptions and except for a translation in the space variable $x$, any global minimizer is radially symmetric