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

created by cupini on 09 May 2014

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Inserted: 9 may 2014
Last Updated: 9 may 2014

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} 1<p\leq q\leq p+1\quad \text{and}\quad q<p\frac{n-1}{n-p}\,. \end{equation}% More regularity applies. Precisely, this solution is also in the class $W_{\text{loc}}^{1,\infty }(\Omega )\cap {W_{\text{loc}}^{2,2}(\Omega )}$.