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 $$\mathrm{d}\mathrm{i}\mathrm{v}[Df(\mathrm{\nabla }u)]=0,$$ 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 $D^{2}f$.