 Uniform power gain for elliptic solutions

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.