  We consider non-negative distributional solutions $u\in C^1 (\bar{B_R } )$ to
the equation $-\mbox{div} [g(|\nabla u|)|\nabla u|^{-1} \nabla u ] = f(|x|,u)$
in a ball $B_R$, with $u=0$ on $\partial B_R $, where $f$ is continuous and
non-increasing in the first variable and $g\in C^1 (0,+\infty )\cap C[0,
+\infty )$, with $g(0)=0$ and $g'(t)>0$ for $t>0$. According to a result of the
first author, the solutions satisfy a certain 'local' type of symmetry. Using
this, we first prove that the solutions are radially symmetric provided that
$f$ satisfies appropriate growth conditions near its zeros. In a second part we
study the autonomous case, $f=f(u)$. The solutions of the equation are critical
points for an associated variation problem. We show under rather mild
conditions that global and local minimizers of the variational problem are
radial.
