 Solution

First replace $u$ by $u+\epsilon$ and later let $\epsilon \downarrow 0$. In the supersolution inequality use the nonnegative test functionwhere $\eta$ is compactly supported. Ellipticity and Young inequality give the Logarithmic Caccioppoli inequalityChoose $\eta =1$ on $B_{\rho }$, supported in $B$, with $|D\eta |\le C(1-\rho {)}^{-1}$. Letting $\epsilon \downarrow 0$ and using Fatou lemma gives