Existence and regularity for elliptic equations under p,q-growth

Under general p,q-growth conditions, we prove that the Dirichlet problem \[\sum_{i=1}^{n}D_{x_{i}}(a^{i}(x,Du))=b(x) \text{in}\,\Omega , u=u_{0} \text{on}\,\partial \Omega\] has a weak solution $u\in W_{loc}^{1,q}(\Omega)$ under the assumptions $1<p\leq q\leq p+1$ and $q<p\frac{n-1}{n-p}$. More regularity applies. Precisely, this solution is also in the class $W_{ loc}^{1,\infty}(\Omega)\cap {W_{loc}^{2,2}(\Omega)}$.