Bernstein's theorem for variational integrals of linear growth and radial structure

We consider entire solutions $u: \mathbb{R}^2 \rightarrow \mathbb{R}$ of the Euler-Lagrange equation associated to the variational integral $\int_{\Omega} g(|\nabla u|)\,dx$ with a strictly convex density $g: [0,\infty)\rightarrow \mathbb{R}$ being of linear growth. We show that the condition $\int_{0}^{\infty} t\,g''(t)\,dt < \infty$ implies the Bernstein property, which means that $u$ must be an affine function. If this condition on g is weakened, we still have some partial Bernstein results.