 Solution

Set $X=Du/\sqrt{1+|Du{|}^2}$. Since $u\in C^2({\mathbb{R}}^n)$, the field is $C^1$, and $|X|\le 1$ everywhere, even when the gradient of $u$ is unbounded. The divergence theorem on $B_R(0)$ givesLetting $R\to \infty$ proves $\kappa =0$. This is the fact that an entire bounded vector field cannot have nonzero constant divergence; it holds in every dimension.

The equation is now exactly the minimal surface equation for a graph. Apply the supplied Bernstein theorem for minimal graphs in dimensions $1\le n\le 7$ to concludeThus no nonzero constant right-hand side is possible for an entire graph, and in the stated dimensions every possible graph is an affine plane.