Variants of Bernstein's theorem for variational integrals with linear and nearly linear growth

Using a Caccioppoli-type inequality involving negative exponents for a directional weight we establish variants of Bernstein's theorem for variational integrals with linear and nearly linear growth. We give some mild conditions for entire solutions of the equation $${\rm div} \Big[Df(\nabla u)\Big] = 0 \, ,$$ under which solutions have to be affine functions. Here $f$ is a smooth energy density satisfying $D^2 f>0$ together with a natural growth condition for $D^2 f$.