Liouville Type Theorems for Elliptic Equations with Gradient Terms

In this paper we obtain Liouville type theorems for nonnegative supersolutions of the elliptic problem ${-Delta u + b(x)|nabla u| = c(x)u}$ in exterior domains of ${mathbb{R}^N}$ . We show that if lim ${{rm inf}_{x longrightarrow infty} 4c(x) - b(x)^2 > 0}$ then no positive supersolutions can exist, provided the coefficients b and c verify a further restriction related to the fundamental solutions of the homogeneous problem. The weights b and c are allowed to be unbounded. As an application, we also consider supersolutions to the problems ${-Delta u + b|x|^{lambda}|{nabla} u| = c|x|^{mu} u^p}$ and ${-Delta u + be^{lambda |x|}|nabla u| = ce^{mu |x|}u^p}$ , where p > 0 and λ, μ ≥ 0, and obtain nonexistence results which are shown to be optimal.