We obtain an a-priori
$$W_{{\mathrm{loc}}}^{1,\infty }\left( \Omega ; {\mathbb {R}}^{m}\right) $$
-bound for weak solutions to the elliptic system
$$\begin{aligned} \text {div}A\left( x,Du\right) =\sum _{i=1}^{n}\frac{\partial }{\partial x_{i}} a_{i}^{\alpha }\left( x,Du\right) =0,\;\;\;\;\;\alpha =1,2,\ldots ,m, \end{aligned}$$
where
$$\Omega $$
is an open set of
$${\mathbb {R}}^{n}$$
,
$$n\ge 2$$
, *u* is a vector-valued map
$$u:\Omega \subset {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{m}$$
. The vector field
$$A\left( x,\xi \right) $$
has a variational nature in the sense that
$$A\left( x,\xi \right) =D_{\xi }f\left( x,\xi \right) $$
, where
$$ f=f\left( x,\xi \right) $$
is a convex function with respect to
$$\xi \in {\mathbb {R}}^{m\times n}$$
. In this context of vector-valued maps and systems, a classical assumption finalized to the *everywhere regularity* is a modulus-dependence in the energy integrand; i.e., we require that
$$f\left( x,\xi \right) =g\left( x,\left| \xi \right| \right) $$
, where
$$ g\left( x,t\right) $$
is convex and increasing with respect to the gradient variable
$$t\in \left[ 0,\infty \right) $$
. We allow *x*-dependence, which turns out to be a relevant difference with respect to the autonomous case and not only a technical perturbation. Our assumptions allow us to consider *both fast and slow growth*. We consider fast growth even of * exponential type*; and slow growth, for instance of *Orlicz-type* with energy-integrands such as
$$g\left( x,\left| Du\right| \right) =a(x)|Du|^{p(x)}\log (1+|Du|)$$
or, when
$$n=2,3$$
, even *asymptotic linear growth* with energy integrals of the type
$$\begin{aligned} \int _{\Omega }g\left( x,\left| Du\right| \right) dx\,=\int _{\Omega }\left\{ \left| Du\right| -a\left( x\right) \sqrt{\left| Du\right| }\right\} dx. \end{aligned}$$