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

Abstract

Abstract We consider the elliptic equation - Δ ⁢ u = u q ⁢ | ∇ ⁡ u | p {-\Delta u=u^{q}|\nabla u|^{p}} in ℝ n {\mathbb{R}^{n}} 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.