= Solution
Use a <Lipschitz continuous> <cutoff function> $\eta$ which equals one on $B_R(x_0)$, vanishes outside $B_{2R}(x_0)$, takes values in $[0,1]$, and has $|\nabla\eta|\leq2/R$. Its <gradient> is supported in the <annulus> $A_R=B_{2R}(x_0)\setminus B_R(x_0)$. The function $\eta^2(u-c)$ is an admissible <test function> in the zero-boundary <Sobolev space> for the <weak solution>, by approximation with smooth compactly supported <test functions>. If $\Lambda_0$ bounds the <operator norm> of $(a^{ij})$, we can take $\Lambda_0=n\Lambda$. The weak equation and <uniform ellipticity> give
$$
\lambda\int\eta^2|\nabla u|^2\leq2\Lambda_0\int_{A_R}\eta|\nabla u|\,|u-c|\,|\nabla\eta|.
$$
<Young inequality> bounds the right side by
$$
\frac\lambda2\int\eta^2|\nabla u|^2+\frac{2\Lambda_0^2}{\lambda}\int_{A_R}|u-c|^2|\nabla\eta|^2.
$$
After absorption, this <annular Caccioppoli inequality> is
$$
\boxed{\int_{B_R(x_0)}|\nabla u|^2\leq\frac{16\Lambda_0^2}{\lambda^2R^2}\int_{A_R}|u-c|^2.}
$$
The dimension is fixed in the notation $C(\lambda,\Lambda)$ of the question; with the entrywise coefficient bound its dependence on $n$ is absorbed there. If the larger ball merely lies in $B$ without its closure being compactly contained, approximation from smaller <cutoff functions> gives the same admissible <test function> and estimate. Crucially, the right side uses only the <annulus>, where the <cutoff function> varies. The constant $c$ is arbitrary because constants have zero <gradient>.
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