$L^p$-estimates (weaker hypothesis) I am reading Chapter 9 of the book of Gilbarg-Trudinger 'Elliptic partial differential equations...'.
I can not understand the following remark:
'We remark also that by invoking the Sobolev imbedding theorem in the proofs of Theorems 9.12 9.13 and 9.14, we may weaken the conditions on the lower order coefficients of L to $b_i\in L^q(\Omega)$ and $c\in L^r(\Omega)$, where $q>n$ if $p\le n$, $q=p$ if $p>n$, $r>n/2$ if $p\le n/2$, $r=p$ if $p>n/2$.
I do not know how I have to reason.
 A: This is a matter of balancing the integrability of the terms; consider an operator of the form
$$ Lu = a^{ij}D_{ij}u + b^iD_iu + cu, $$
using the summation convention. Suppose a-priori that $u \in W^{2,p}(\Omega),$ and assume first that $p < n/2.$ Then by Sobolev embedding we have
$$ Du \in L^{\frac{np}{n-p}}, \quad u \in L^{\frac{np}{n-2p}}. $$
Hence we have $b^i D_iu \in L^p$ provided $b^i \in L^n$ and similarly $cu \in L^p$ provided $c \in L^{\frac n2}.$ Hence the equation remains balanced with these worse regularity assumptions of the lower-order terms, so informally we expect $L^p$ estimates to also hold in this setting.
To make this precise we need to prove the a-priori estimate in Theorem 9.11 under the weaker assumptions, from which the rest will follow. I will be quite light on the details, but hopefully you will see how to fill in the details from the below sketch. The main step we need to amend is the estimate
$$ \lVert \eta a^{ij}D_{ij}u +2 a^{ij}D_i\eta D_ju + u a^{ij}D_{ij}\eta\Vert_{p;B_R} \leq C \left( \lVert f \rVert_{p;B_R} + \frac1{(1-\theta)R} \lVert Du \rVert_{p;B_{\sigma' R}} + \frac1{(1-\theta)^2R^2}\lVert u \rVert_{p;B_R}\right).$$
This I have copied from the bottom of p236 in my copy. The point here is that you have bounds in terms of $a^{ij}D_{ij}u,$ which you want to write in terms of $f = Lu$ and lower order terms. That is, to estimate
$$ \lVert \eta a^{ij}D_{ij}u \rVert_{p;B_R} \leq \lVert f \rVert_{p;B_R} + \lVert \eta(b^iD_iu + cu) \rVert_{p;B_R}. $$
This is where we need the idea of balancing terms. By using a scale-invariant version of the Sobolev embedding (or something similar, the below is not quite correct as written), we can estimate
$$ \lVert b^iD_iu \rVert_{p;B_R} \leq C\lVert b^i \rVert_{n;B_R} \lVert D^2u \rVert_{p;B_R} \leq CR^{\delta(n,p,q)} \lVert b^i \rVert_{q;B_R} \lVert D^2u\rVert_{p;B_R}, $$
and similarly for the $c$ term we also get $\lVert cu \rVert_{p;B_R} \leq C R^{\tilde\delta(n,p,r)} \lVert c \rVert_{r;B_R} \lVert D^2u \rVert_{p;B_R}.$ Then the idea is that for $R$ small enough we can absorb these second order terms to the LHS, similarly to how the proof is concluded.
If $p > n/2$ then the estimate for the $cu$ term simplifies because $u$ is bounded, and hence we can take $r=p.$ The same holds for when $p > n$ and also the borderline cases should follow with some care.
