Power Caccioppoli inequality ($\int |Du{|}^2{u}^{\beta -2}{\eta }^2\le \frac{C}{(\beta -1{)}^2}\int u^{\beta }|D\eta {|}^2$)

For a nonnegative bounded weak subsolution and $\beta >1$, test with ${\eta }^2(u+\epsilon {)}^{\beta -1}$. Uniform ellipticity and Cauchy-Schwarz inequality give $\int |Du{|}^2{u}^{\beta -2}{\eta }^2\le 4{\Lambda }_{\ast }^2/[{\lambda }^2(\beta -1{)}^2]\int u^{\beta }|D\eta {|}^2$, with ${\Lambda }_{\ast }$ an operator norm bound for the coefficient matrix. The Sobolev chain rule and $\epsilon \downarrow 0$ justify fractional powers and zero values. An entrywise bound $\Lambda$ yields ${\Lambda }_{\ast }\le n\Lambda$.

For a nonnegative $H^1$ weak solution of a homogeneous divergence-form elliptic equation on $B_1\subset {\mathbb{R}}^{\ell }$, $\ell \ge 2$, with symmetric measurable $\lambda I\le A\le \Lambda I$, set $R=\Lambda /\lambda$ and $\alpha =1+1/\ell$. There is $D_{\ell }\ge 1$ independent of $q\ge 2$ such thatTest the equation with a cutoff times $u^{q-1}$, first regularized or truncated, to get $\int {\zeta }^2|\nabla u^{q/2}{|}^2\le [q/(q-1){]}^2{R}^2\int u^q|\nabla \zeta {|}^2$. Apply a fixed-domain Sobolev embedding theorem to $\zeta u^{q/2}$ and take the $2/q$ power. Keeping the constant inside that power is crucial for Moser iteration.