 Solution

NormalizeThen $0\le v_k\le 1$, 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 $\rho <1$,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 $B$.