= Solution
Normalize
$$
v_k=\frac{u_k}{\sup_Bu_k}.
$$
Then $0\leq v_k\leq1$, and $v_k$ remains a weak supersolution. The ordinary <Caccioppoli inequality> with a cutoff supported in $B_R$ gives a uniform $W^{1,2}(B_\rho)$ bound whenever $\rho<R<1$. The <Rellich-Kondrachov compactness theorem> and a diagonal subsequence therefore give, for every $\rho<1$,
$$
v_{k'}\to v\quad\text{strongly in }L^2(B_\rho),
\qquad
v_{k'}\rightharpoonup v\quad\text{weakly in }W^{1,2}(B_\rho).
$$
The strong convergence preserves nonnegativity. Passing to the limit in the linear supersolution inequality against every nonnegative compactly supported test function shows that $v$ is a weak supersolution in $B$.
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