  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 $D^2 f$.
