= Solution
For the <divergence-form elliptic operator> use the <weak subsolution> convention $\int a^{ij}D_juD_i\varphi\leq0$ for nonnegative <test functions>. Set $u_\varepsilon=u+\varepsilon$ and test with $\varphi=\eta^2u_\varepsilon^{\beta-1}$, where $\beta>1$. The <Sobolev chain rule> makes this test admissible; boundedness and the positive regularization control the derivative of its power. Ellipticity and the <Cauchy-Schwarz inequality> give, with $X=\int|Du|^2u_\varepsilon^{\beta-2}\eta^2$ and $Y=\int u_\varepsilon^\beta|D\eta|^2$,
$$
\lambda(\beta-1)X\leq2\Lambda_*X^{1/2}Y^{1/2},\qquad
\Lambda_*\geq\operatorname*{ess\,sup}\|a(x)\|_{\mathrm{op}}.
$$
Division when $X>0$, and the trivial case $X=0$, establish the <power Caccioppoli inequality>. Letting $\varepsilon\downarrow0$ gives
$$
\boxed{\int|Du|^2u^{\beta-2}\eta^2\leq
\frac{4\Lambda_*^2}{\lambda^2(\beta-1)^2}\int u^\beta|D\eta|^2.}
$$
For $1<\beta<2$ this limit uses the <Fatou lemma>; the <gradient of a Sobolev function vanishes on a level set>, so the weighted integrand is assigned zero on $\{u=0\}$. The same regularization justifies differentiating $u^{\beta/2}\eta$ in $W^{1,2}$.
The PDF bounds individual entries by $\Lambda$. Then $\Lambda_*\leq n\Lambda$: dimensional factors are included in the ellipticity constants when the first estimate is written as $C(\lambda,\Lambda)$. With literal entrywise bounds, the explicit valid constant displayed here is $4n^2\Lambda^2/\lambda^2$. If $\Lambda$ bounds the <operator norm> instead, it is $4\Lambda^2/\lambda^2$. This distinction does not alter any later conclusion, whose constants explicitly depend on $n$.
For $n\geq3$, put $\sigma=n/(n-2)$. Apply the <Sobolev inequality> to $g=u^{\beta/2}\eta$ and use
$$
\int|Dg|^2\leq\frac{\beta^2}{2}\int|Du|^2u^{\beta-2}\eta^2
+2\int u^\beta|D\eta|^2.
$$
For $\beta\geq p>1$, $\beta/(\beta-1)\leq p/(p-1)$, so the resulting constant is uniform in $\beta$. Taking a cutoff equal to one on $B_r$, supported in $B_R$, with $|D\eta|\leq C/(R-r)$, yields
$$
\boxed{\|u\|_{L^{\sigma\beta}(B_r)}
\leq C_p^{1/\beta}(R-r)^{-2/\beta}\|u\|_{L^\beta(B_R)},\qquad\beta\geq p.}
$$
At $R=1$ a compactly supported cutoff can still be chosen with this bound by leaving an outer margin; alternatively pass to the limit from smaller radii.
For <Moser iteration>, set $\beta_j=p\sigma^j$ and $r_j=1/2+2^{-j-1}$. Repeated use of the preceding estimate bounds the successive norms by
$$
\|u\|_{L^p(B_1)}\prod_{j\geq0}C_p^{1/\beta_j}2^{(2j+4)/\beta_j}.
$$
This product is finite because
$$
\sum_{j\geq0}\frac1{\beta_j}=\frac n{2p},\qquad
\sum_{j\geq0}\frac j{\beta_j}=\frac{n(n-2)}{4p}.
$$
As the exponents tend to infinity, their norms on the <Euclidean ball> of radius $1/2$ tend to its <essential supremum>. Therefore
$$
\boxed{\|u\|_{L^\infty(B_{1/2})}\leq C_{n,p,\lambda,\Lambda}\|u\|_{L^p(B_1)}.}
$$
The power $\beta$ is independent of the exponents of <Hölder continuity> in earlier questions.
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