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Put $\alpha =1-n/q>0$ andFirst suppose $A<\infty$. For an interior ball $B_R$, with $R\le 1$, write $M_R={\mathrm{esssup}}_{B_R}u$, $m_R={\mathrm{essinf}}_{B_R}u$, and $\omega (R)=M_R-m_R$. Constant shifts of a divergence-form elliptic operator change its forcing. The two nonnegative weak supersolutions $v=M_R-u$ and $w=u-m_R$ satisfy, respectively,Their Weak Harnack inequalities on $B_{R/2}$ therefore have forcing errors bounded by $CN{R}^{\alpha }$. Indeed $|M_R|,|m_R|\le A$, the new divergence forcing is bounded by $\Vert F{\Vert }_q+A\Vert b{\Vert }_q$, and the scalar forcing by $\Vert g{\Vert }_{q/2}+A\Vert d{\Vert }_{q/2}$; also $R^{2\alpha }\le R^{\alpha }$.

On $B_{R/2}$ we have $v+w=\omega (R)$. Even if $p<1$, the elementary inequality $(s+t{)}^p\le C_p(s^p+t^p)$ impliesThe sum of the two essential infima is $\omega (R)-\omega (R/2)$. Enlarging a uniform constant $C_0>1$ giveswhere $\theta =1-C_0^{-1}\in (0,1)$. This is an oscillation decay estimate; inhomogeneous oscillation decay implies Hölder continuity. ChooseIterating on concentric balls, starting at a fixed admissible radius $R_{\ast }\le 1$, yieldsThe strict choice of $\mu$ bounds the geometric series, including the case in which the two original decay rates coincide. Monotonicity of oscillation extends this bound to intermediate radii.

Take $R_{\ast }$ uniformly smaller than the distance of ${\Omega }^{\prime }$ from the boundary of $\Omega$. The vanishing essential oscillation produces a continuous representative of $u$. A ball centred at either of two sufficiently close points of ${\Omega }^{\prime }$ contains both, so its oscillation bounds their difference by $CN|x-y{|}^{\mu }$. For separated points use $|u(x)-u(y)|\le 2A$. Thus the Hölder seminorm satisfiesThe constants depend only on the indicated domain separation, dimension, coefficient norms, uniform ellipticity bounds and $q$. If the global essential supremum is infinite, the displayed global estimate is vacuous; local boundedness of weak elliptic subsolutions on slightly larger compact subsets still makes the same argument prove interior Hölder continuity.

Let $M={\mathrm{esssup}}_{\Omega }u$. 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, $v\ge 0$ and $Lv=-Lu\le 0$. The hypothesis says ${\mathrm{essinf}}_{B}v=0$. 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 givesConsequently $v=0$ 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 $u=M$ 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 $L(M-u)=-Lu$.