= Solution
Use the interior $\Omega=\{1/2<|x|<2\}$ for the <Sobolev space> notation; the closed shell in the PDF specifies its boundary. Put $A(x)=\operatorname{diag}(1,1,g(|x|))$. A <weak solution> has zero boundary trace in the sense of the <Sobolev trace theorem>, equivalently $u\in H_0^1(\Omega)$, and satisfies
$$
\boxed{\int_\Omega\bigl(u_1\varphi_1+u_2\varphi_2+g(|x|)u_3\varphi_3\bigr)\,dx=-\int_\Omega f\varphi\,dx\quad(\varphi\in H_0^1(\Omega)).}
$$
The minus sign is necessary because the operator in the paper is positive divergence, not its negative. One can first test with $C_c^\infty$ functions and then use density in the <zero-boundary Sobolev space>. Smoothness of $g$ makes its restriction to the compact radial interval bounded, so both sides define continuous functionals on this space. Merely belonging to $H^1$ without the zero trace would omit the <boundary condition>.
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