Let $B=B_{\exp(-2)}\subset\mathbb{R}^2$.
I would like to show that a weak solution to the following system (in $B$): \begin{align*} \triangle u_1&=-2|Du|^2(u_1+u_2)/(1+|u|^2)\\ \triangle u_2&=2|Du|^2(u_1-u_2)/(1+|u|^2) \end{align*} is \begin{equation*} u_1=\sin \big(\log (\log |x|^{-1})\big), \quad u_2=\cos \big(\log (\log |x|^{-1})\big). \end{equation*}I have already shown that $u=(u_1, u_2)\in L^{\infty}(B)\cap H^1(B)$. Further, $u$ has certain second weak derivatives in $B$. Also, \begin{equation} u\in L^{\infty}(B)\cap H^1(B)\cap C^{\infty}(B\setminus\{0\}) \end{equation}and $u$ satisfies the system outside the origin (I have shown this).
To show that it is a weak solution, I would like to show that for any $\phi\in L^{\infty}(B)\cap H^1(B)$ with compact support in $B$ I have
\begin{equation} -\int_BDu_i\cdot D\phi\ \mathrm{d}x=\int_B \triangle u_i\phi\ \mathrm{d}x \end{equation}for each $i=1, 2$.
I started off by noting that if $\phi\in L^{\infty}(B)\cap H^1(B)$ with compact support in $B$, then $T\phi=0$ where $T$ is the trace operator, hence $\phi\in H_0^1(B)=\overline{C_c^{\infty}(B^0)}$. Consequently, there exists a sequence, $\{\phi_m\}_{m=1}^{\infty}\subset C_c^{\infty}(B^0),$ such that \begin{equation} \|\phi_m-\phi\|_{H^1(B)}\rightarrow 0\quad(\text{as }m\rightarrow \infty). \end{equation}This gives us \begin{equation} -\int_BDu_i\cdot D\phi_m\ \mathrm{d}x=\int_B \triangle u_i\phi_m\ \mathrm{d}x \end{equation}since $u_i$ has second weak derivatives (in those directions). Then \begin{equation} \|Du_i(D\phi_m-D\phi)\|_{L^1(B)}\leq\|Du\|_{L^2(B)}\|D\phi_m-D\phi\|_{L^2(B)}\rightarrow 0 \end{equation}as $m\rightarrow\infty$. Therefore, the left hand side converges to \begin{equation} -\int_BDu_i\cdot D\phi\ \mathrm{d}x \end{equation} for each $i=1, 2$ and now I want to show that the right hand side converges to \begin{equation} \int_B\triangle u_i\phi\ \mathrm{d}x \end{equation}for each $i=1, 2$. This is where I am having some difficulty. I thought if I could show that the operator $L_i: H_0^1(B^0)\rightarrow\mathbb{R}$ defined for each $i=1, 2$ as \begin{equation} L_i\phi\equiv\int_B \triangle u_i\phi\ \mathrm{d}x, \end{equation}is a bounded linear operator then by the density of $C_c^{\infty}(B^0)$ in $H_0^1(B)$ I would have $L_i\phi_m\rightarrow L_i\phi$, as desired. However, in trying to show that $L_i$ is bounded, I get as far as \begin{equation} \int_{B}|\triangle u_i||\phi_m-\phi|\ \mathrm{d}x\leq C\int_B|Du|^2|\phi_m-\phi|\ \mathrm{d}x \end{equation}and then I'm not sure how to proceed from here.