  We prove a Bernstein theorem for $\Phi$-anisotropic minimal hypersurfaces in
all dimensional Euclidean spaces that the only entire smooth solutions $u:
\mathbb{R}^{n}\rightarrow \mathbb{R}$ of $\Phi$-anisotropic minimal
hypersurfaces equation are linear functions provided the anisotropic area
functional integrand $\Phi$ is sufficiently $C^{3}$-close to classical area
functional integrand and $|\nabla u(x)|=o(|x|^{\varepsilon})$ for
$\varepsilon\leq \varepsilon_{0}(n, \Phi)$ with the constant
$\varepsilon_{0}(n, \Phi)>0$.
