This is exercise 10.11 from Leoni's book.
Take $\Omega:=\{x\in \mathbb R^N;\,\,|x|>1\}$ and let $$u(x):=1-|x|^{2-N}$$ for $N\geq 3$.
I am trying to prove that $\frac{\partial^2u}{\partial x_i\partial x_j}\in L^{p}(\Omega)$ for all $1\leq p<\infty$. Clearly, I have $$\frac{\partial^2u}{\partial x_i\partial x_j} = (N-2)\frac{\delta_{ij}|x|^N-Nx_ix_j|x|^{N-2}}{|x|^{2N}} $$ Suppose $i\neq j$, I have $$\int_{\Omega}\left|\frac{\partial^2u}{\partial x_i\partial x_j}\right|^p\leq \int_\Omega \frac{1}{|x|^{Np}}dx = \int_1^\infty r^{-Np+N-1}dr$$ To have integration convergen, I need $-Np+N-1<-1$ which is $p>1$...
I have no idea about how to prove the case $p=1$.
Also, when $i=j$, I just don't know how to deal with $$ (N-2)\frac{|x|^N-Nx_i^2|x|^{N-2}}{|x|^{2N}} $$
Any help is really welcome!
The exercise is incorrectly stated: the second derivatives of $u$ are in $L^p$ for $1<p<\infty$ (in fact, also for $p=\infty$), but not for $p=1$.
The case $i=j$ is not much different from $i\ne j$: just use the triangle inequality to estimate the derivative by $C|x|^{-N}$.
Remark: disregarding $1$, we have a function $|x|^{2-N}$, which is homogeneous of degree $2-N$. Each derivative of order $k$ is homogeneous of degree $2-N-k$. This is already enough to determine its degree of integrability. For a function $\phi$ homogeneous of degree $d$, the integral of $|\phi|^p$ over dyadic shell $2^m<|x| \le 2^{m+1}$ is $C 2^{mpd} 2^{mN}$, and the series $\sum_{m\ge 0} 2^{mpd} 2^{mN}$ converges iff $pd+N<0$.