Let $u \in H^1_{loc}(\Omega)$ satisfy, for some $c$, the following reverse Sobolev inequality, also known as Caccioppoli inequality: $$\int_{B_\rho \cap \{u>k\}}|\nabla u|^2\leq \frac{c}{(R-\rho)^2}\int_{B_R \cap \{u>k\}}(u-k)^2$$
for any $B_\rho \subset \subset B_R \subset \subset \Omega$. It follows that also $(u-k)_+\in H^1_{loc}$, and by application of the Sobolev inequality, $(u-k)_+ \in L^{2^*}_{loc}$, with the estimate:
$$||(u-k)_+||_{L^{2^*}(B_\rho)}^2\leq C\int_{B_\rho}|\nabla (u-k)_+|^2=C\int_{B_\rho \cap \{u>k\}}|\nabla u|^2$$
Combine with the first inequality and one has $$||u-k||_{L^{2^*}(B_\rho\cap \{u>k\})}^2 \leq \gamma \frac{||u-k||_{L^{2}(B_R\cap \{u>k\})}^2}{(R-\rho)^2}$$
Was there anything wrong in the derivation of such estimate? This would be a simplification of the proof of Thm. 3.41 in this book. The context is the De Giorgi's program to the 19th Hilbert's problem.
There is an issue with applying the Sobolev inequality $$ \lVert (u-k)_+ \rVert^2_{L^{2^*}(B_{\rho})} \,\mathrm{d}x\leq C \int_{B_{\rho}} |\nabla(u-k)_+|^2 \,\mathrm{d}x, $$ which typically requires either $(u-k)_+$ to vanish somewhere, or to have zero average. Else by considering $u \equiv k_1 > k$ we see this cannot hold in general.
For this reason, one needs to consider $\eta\,(u-k)_+$ for a sutiable cutoff function $\eta$ instead. Otherwise the proof appears to be largely the same to me, just that they are keeping more careful track of constants which presumably will be useful in the later iteration arguments.