*Published Paper*

**Inserted:** 9 may 2014

**Last Updated:** 9 may 2014

**Journal:** Adv. Differential Equations

**Volume:** 19

**Pages:** 693–724

**Year:** 2014

**Abstract:**

Under general $p,q$-growth conditions, we prove that the Dirichlet problem
\begin{equation**}
\left\{
\begin{array}{ll}
\sum _{{i=1}}^{{n}\frac{\partial} }{\partial x_{{i}}a}^{{i}}(x,Du)=b(x) & \quad
\text{in}\,\Omega , \\
u=u_{{0}} & \quad \text{on}\,\partial \Omega%
\end{array}%
\right.
\end{equation**}
has a weak solution $u\in W_{\mathrm{loc}}^{1,q}\left( \Omega \right) $
under the assumptions
\begin{equation

**Keywords:**
Lipschitz regularity, Elliptic equation, existence of solutions, $p,q$-growth conditions

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