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

$$\begin{aligned} {\text {div}} \Big [Df(\nabla u)\Big ] = 0 \,, \end{aligned}$$

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 \(D^2 f\).