  We consider the elliptic equation $-\Delta u = u^q|\nabla u|^p$ in $\mathbb
R^n$ for any $p\ge 2$ and $q>0$.
  We prove a Liouville-type theorem, which asserts that any positive bounded
solution is constant.
  The proof technique is based on monotonicity properties for the spherical
averages of sub- and super-harmonic functions, combined with a gradient bound
obtained by a local Bernstein argument.
  This solves, in the case of bounded solutions,
  a problem left open in~\cite{BVGHV}, where the authors consider the case
$0<p<2$. Some extensions to elliptic systems are also given.
