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 \[ { div} [Df(∇ u)] = 0 \, , \] under which solutions have to be affine functions. Here f is a smooth energy density satisfying D² f>0 together with a natural growth condition for D² f.