*Accepted Paper*

**Inserted:** 22 dec 2014

**Last Updated:** 22 dec 2014

**Journal:** Ann. Acad. Sci. Fenn. Math.

**Year:** 2015

**Abstract:**

We consider a non-linear system of $m$ equations in divergence form and a boundary condition:
\[
\left\{\begin{array}{cl}
\displaystyle
\sum_{i=1}^n\frac{\partial }{\partial x_i}\left(A_i^\alpha(x,Du(x))\right)=0, \quad 1\le \alpha\le m,
&\text{in $ \Omega$}
\\
u=\tilde{u} &\text{on $ \partial \Omega$.}
\end{array}\right.
\]
The functions $A_i^\alpha(x,z)$ are H\"older continuous with respect to $x$ and
\[

z

^p-c_1\le \sum_{\alpha=1}^m\sum_{i=1}^n
A_i^{\alpha}(x,z)z_i^{\alpha}\le c_2 (1+

z

)^{q},\qquad 2\le p\le q.
\]
We prove the existence of a weak solution $u$ in $(\tilde{u}+W_0^{1,p}(\Omega;\mathbb{R}^m))\cap W_{\rm loc}^{1,q}(\Omega;\mathbb{R}^m)$,
provided $p$ and $q$ are close enough and under suitable summability assumptions on
the boundary datum $\tilde{u}$.

**Keywords:**
existence, regularity, weak solution, elliptic system, growth

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