Consider the system \(\begin{aligned} \left\{ \begin{aligned}&\Delta u \ge p(x)g(v) \quad & \text{ in } \mathbb {R}^n, \\&\Delta v \ge q(x)f(|\nabla u|) \quad & \text{ in } \mathbb {R}^n, \\&u>0, v>0 \quad & \text{ in } \mathbb {R}^n, \end{aligned} \right. \end{aligned}\) where \(n\ge 2\) , \(f,g\in C[0,\infty )\) and \(p,q\in C(\mathbb {R}^n)\) . If f, g are non-decreasing and convex, using a comparison argument we prove a sufficient Keller-Osserman type condition for the nonexistence of entire solutions. This result is sharp for a wide class of systems. Similar results are also shown to hold for systems without gradient terms.
