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

Abstract

Under general $p,q$-growth conditions, we prove that the Dirichlet problem \begin{equation*} \left\{ \begin{array}{ll} \displaystyle \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} \Big ( \Omega \Big ) $ under the assumptions $ 1 < p\leq q\leq p+1$ and $q < p\tfrac{n-1}{n-p}. $ More regularity applies. Precisely, this solution is also in the class $W_{ \text{loc}}^{1,\infty }(\Omega )\cap {W_{\text{loc}}^{2,2}(\Omega )}$.