A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient

We consider the elliptic equation -Delta u = u^q|nabla u|^p in R^n for any p>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 a recent paper, where the authors consider the case 0<2. Some extensions to elliptic systems are also given.