# Doubt about the proof of Moser Iteration in Gilbarg & Trudinger's book

I was reading Theorem 8.15 about Moser Iteration in Gilbarg and Trudinger's monograph. I understand all the steps of the given proof, but I have the following doubts which could not be cleared by a careful reading.

1. The Authors, as hypotheses for the theorem, require that $$f^i\in L^q(\Omega)$$, $$i=1,\ldots,n$$ and $$g\in L^{q/2}(\Omega)$$ for some $$q>n$$ but it seems they haven't used these facts anywhere in the proof: is this so and, if not, in which steps are these facts used?

2. Does the theorem fail for $$q\le n$$?

Here I have uploaded a snapshot of the theorem.

Equation 8.3

$$$$Lu=D_i(a^{ij}(x)D_ju+b^i(x)u)+c^i(x)D_iu+d(x)u$$$$.

Equation 8.30

$$$$\int_{\Omega}\left(D_ivA^i-vB\right)dx=(\le,\ge)0$$$$

Equation 8.32

$$$$\bar z=|z|+k,\qquad \bar b=\lambda^{-2}(|b|^2+|c|^2+k^{-2}|f|^2)+\lambda^{-1}(|d|+k^{-1}|g|)$$$$

Equation 8.33

\begin{align} p_iA^i(x,z,p) & \ge \frac{\lambda}{2}(|p|^2-2\bar b\bar z^2) \\ | \bar zB(x,z,p) | &\le \frac{\lambda}{2}\left( \epsilon|p|^2+\frac{\bar b}{\epsilon}\bar z^2\right) \end{align}

Any Help Hint will be greatly appreciated

• The proof references many inequalities (or other stuff, such as (8.32) ) which are not included in the question. Do some of them depend on $f^i$, $g$? Aug 19 '20 at 18:47
• Maybe before (8.36): choosing $k$ as in the statement of the theorem. What is $\bar{b}$? Aug 20 '20 at 9:38
it definitely needs the condition $$f^i\in L^q(\Omega)$$ and $$g\in L^{q/2}(\Omega)$$.
1. During the proof, one needs to choose $$\chi=\hat{n}(q-2) / q(\hat{n}-2)>1$$ (above equation (8.37)). This is possible if and only if $$q>\hat n$$.
2. The theorem in general fails for $$q\leq n$$. One can get some clue from the $$W^{2,p}$$ estimates of elliptic equations. Conside a special case, $$f=0$$ and $$Lu=g$$ with $$u=0$$ on the boundary. The $$W^{2,p}$$ roughly says $$||u||_{W^{2,q/2}}\leq C||g||_{L^{q/2}}$$ Recall the Sobolev embedding theorem, $$W^{2,q/2}\in L^\infty$$ if $$q>n$$, while this is not true when $$q\leq n$$.
For a counterexample, one can just take one element $$g\in W^{2,n/2}$$ but not in $$g\not\in L^\infty(\Omega)$$. Then $$\Delta u=\Delta g$$ has a solution $$u$$ while (8.34) can not be true.