  We study nonlinear elliptic equations in divergence form
$${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When
${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal
A}(x,\xi)$ enjoys $B^\alpha_{\frac{n}\alpha, q}$ smoothness, local
well-posedness is found in $B^\alpha_{p,q}$ for certain values of
$p\in[2,\frac{n}{\alpha})$ and $q\in[1,\infty]$. In the particular case
${\mathcal A}(x,\xi)=A(x)\xi$, $G=0$ and $A\in B^\alpha_{\frac{n}\alpha,q}$,
$1\leq q\leq\infty$, we obtain $Du\in B^\alpha_{p,q}$ for each
$p<\frac{n}\alpha$. Our main tool in the proof is a more general result, that
holds also if ${\mathcal A}$ has growth $s-1$ in $Du$, $2\leq s\leq n$, and
asserts local well-posedness in $L^q$ for each $q>s$, provided that
$x\mapsto{\mathcal A}(x,\xi)$ satisfies a locally uniform $VMO$ condition.
