= Solution
First prove the hinted <Caccioppoli inequality>. Testing the <weak formulation> with $u_k\varphi^2$ is legitimate by approximation in the zero-boundary <Sobolev space>. Let $\Lambda_*=\operatorname*{ess\,sup}\|a(x)\|_{\mathrm{op}}\leq n\Lambda$. Then
$$
\lambda\int|Du_k|^2\varphi^2
\leq2\Lambda_*\int|u_k|\,|Du_k|\,|\varphi|\,|D\varphi|
\leq\tfrac\lambda2\int|Du_k|^2\varphi^2
+\frac{2\Lambda_*^2}{\lambda}\int u_k^2|D\varphi|^2,
$$
by <Young inequality>. Hence
$$
\boxed{\int|Du_k|^2\varphi^2\leq\frac{4\Lambda_*^2}{\lambda^2}\int u_k^2|D\varphi|^2.}
$$
No derivative of the measurable coefficient <matrix> is taken.
Set $w_k=u_k/\sup_{B_1}|u_k|$. The denominator is finite and positive because $u_k$ is continuous on the compact closure and nonzero. Each $w_k$ solves the same linear equation and $|w_k|\leq1$. For every $\theta<1$, choose a <cutoff function> equal to one on $B_\theta$ and supported in a slightly larger interior <Euclidean ball>. The energy bound gives a uniform $W^{1,2}(B_\theta)$ bound.
Take radii $\theta_j\uparrow1$. Weak compactness and a <diagonal subsequence argument> produce one subsequence converging weakly in $W^{1,2}(B_{\theta_j})$ for every $j$, with consistent restrictions defining $v\in W^{1,2}_{\mathrm{loc}}(B_1)$. Part (b) and the <Arzelà-Ascoli theorem> allow a further subsequence converging uniformly on $\overline{B_{1/4}}$. Its uniform limit agrees almost everywhere with the weak limit and supplies its continuous representative there.
For every smooth compactly supported <test function>, choose $j$ containing its support. Since $a^{ij}D_i\varphi\in L^2$ and $Dw_k$ converges weakly there,
$$
\int a^{ij}D_jvD_i\varphi
=\lim_k\int a^{ij}D_jw_kD_i\varphi=0.
$$
Thus the <compactness of normalized weak elliptic solutions> gives
$$
\boxed{w_{k'}\longrightarrow v\text{ uniformly on }\overline{B_{1/4}},\qquad
Lv=0\text{ weakly in every }B_\theta,\ 0<\theta<1.}
$$
The diagonal step is needed to retain the equation beyond the small <Euclidean ball> of <uniform convergence>. The limit need not be nonzero: for the <Laplacian> in dimension at least two, the normalized <harmonic functions> $\operatorname{Re}(x_1+ix_2)^k$ tend uniformly to zero on each smaller <Euclidean ball>.
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