Boundedness and differentiability for nonlinear elliptic systems

We consider the elliptic system $\operatorname{div} (\mathcal{A}^j (x,u,\nabla u)) = \mathcal{B}^j (x,u,\nabla u)$, $j=1,\ldots,N,$and an obstacle problem for a similar system of variational inequalities. The functions $\mathcal{A}^j$ and $\mathcal{B}^j$ satisfy certain ellipticity and boundedness conditions with a $p$-admissible weight $w$ and exponent $1<p\le2$. The growth of $\mathcal{B}^j$ in $\vert\nabla u\vert$ and $\vert u\vert$ is of order $p-1$. We show that weak solutions of the above systems are locally bounded and differentiable almost everywhere in the classical sense.