A Liouville-Type Theorem for an Elliptic Equation with Superquadratic Growth in the Gradient

We consider the elliptic equation -Δ⁢u=uq⁢|∇⁡u|p{-Δ u=uq|∇ u|ᵖ} in ℝn{Rⁿ} for any p>2{p>2} and q>0{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 [2], where the case 0<p<2{0<p<2} is considered. Some extensions to elliptic systems are also given.