I have an open bounded set $\Omega$ with smooth boundary $\partial\Omega$ and I would like to prove that a certain function $u^{-\gamma}$, with $u:\mathbb{R}^n\rightarrow \mathbb{R}$ and $\gamma>0$ is in $L^p(\Omega)$ with $p>n$ and $\gamma$ small enough.
I was able to show that $u$ satisfies $r_1\delta_{\Omega}(x)<u(x)<r_2\delta_\Omega(x)$, where $r_1$, $r_2$ are positive constants and $\delta_\Omega(x)=d(x,\partial \Omega)$ is the "distance to the boundary" function. Therefore, It would be sufficcient to prove that $(\delta\Omega)^{-\gamma}$ is in $L^p(\Omega)$ with $p>n$ for $\gamma$ small enough.
It is easy to show that in the case $\Omega = B(0,M)$ (ball of radius $M$) this last statement is true. All you have to do is integrate
\begin{equation} \int_0^M \int_{\partial B(0,s)} (\delta_{\Omega})^{-\gamma p}d\mu ds= k\int_0^M \frac{s^{n-1}}{(M-s)^{\gamma p}} ds \leq \\ k M^{n-1} \int_0^M \frac{1}{(M-s)^{\gamma p}} ds\leq k' \int_0^M \frac{1}{(r)^{\gamma p}} dr \end{equation}
and for any fixed $p$ this last integral is finite if we choose $\gamma$ small enough. Therefore, I tend to believe that for other domains that are not too irregular the same thing may happen but could not come up with a proof.
So the question is if $(\delta\Omega)^{-\gamma}$ is in $L^p(\Omega)$ with $p>n$ for $\gamma$ small enough. And as a plus, which hypothesis should I ask about the set $\partial \Omega$ for this claim to be true? Is it necessary to be smooth or can be just $C^2$ for example, do I need any other special property besides the smothness ones?