= Solution
Let $M=\operatorname*{ess\,sup}_{\Omega}u$. The standard <local boundedness of weak elliptic subsolutions> gives a finite upper bound on every <relatively compact> ball. In particular, the assumed equality of <essential suprema> makes $M$ finite. Equivalently, when using the maximum principle for an upper-bounded <weak subsolution>, this preliminary fact is already part of the hypothesis.
Set $v=M-u$. Because the <divergence-form elliptic operator> here annihilates constants, $v\geq0$ and $Lv=-Lu\leq0$. The hypothesis says $\operatorname*{ess\,inf}_Bv=0$. Cover the compact closure of $B$ by finitely many balls $D_j$ whose doubled balls lie in $\Omega$. At least one $D_j$ has <essential infimum> zero: otherwise the minimum of their finitely many positive lower bounds would be a positive lower bound on $B$. The <Weak Harnack inequality> with zero forcing now gives
$$
\left(\frac{1}{|D_j|}\int_{D_j}v^p\right)^{1/p}\leq C\operatorname*{ess\,inf}_{D_j}v=0.
$$
Consequently $v=0$ almost everywhere on $D_j$. If another admissible ball overlaps $D_j$ in an open set, its <essential infimum> of $v$ is also zero, and the same argument makes $v$ vanish there. Since a domain is <connected>, a chain of overlapping interior balls reaches every point of $\Omega$. This <propagation of zeros by the weak Harnack inequality> proves \b[$u=M$ almost everywhere on $\Omega$], which is the claimed <strong maximum principle>. Notice why the absence of the $b,d$ terms matters: it is precisely what makes $L(M-u)=-Lu$.
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