Pathological solutions to elliptic problems in divergence form with continuous coefficients

We construct a function $u\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}(B(0,1))$ which is a solution to $\text{Div}(A\mathrm{\nabla }u)=0$ in the sense of distributions, where $A$ is continuous and $u\in ̸W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(B(0,1))$ for $p>1$. We also give a function $u\in W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,1}(B(0,1))$ such that $u \in W^{1,p}{$\mathrm{l}\mathrm{o}\mathrm{c}$}(B(0,1))$forevery$p < \infty$,$u$satisfies$\Div (A \nabla u)=0$with$A$continuousbut$u \not \in W^{1, \infty}{$\mathrm{l}\mathrm{o}\mathrm{c}$}(B(0,1))$.