I am looking for a reference on the following problem (or related problems):
If you have $\forall \varphi\in W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)$ and $w\in L^{p'}$ (where $\frac{1}{p} + \frac{1}{p'} =1 $):
$$ 0 = \int\limits_\Omega w (-\Delta\varphi + f\varphi), $$
i.e. $w$ is a "very weak solution" of
\begin{align*} -\Delta w + fw & = 0 \text{ in } \Omega\\ w & =0 \text{ on } \partial\Omega. \end{align*}
where $f$ is continuous (maybe more). Do you know any reference that in this case, it is indead $w\in W^{2,p}(\Omega)\cap W^{1,p}_0(\Omega)$ ? I looked at various other posts, but didn't find a reference for this case. Thank you.
I found a reference that maybe can help you. It's
in particular chapter 3, Theorem 3.5 is close to the result you want with $\Omega=\mathbb R^n$, however it's not exactly that. I'll try to briefly summarize the result here. Consider the equation $$\Delta w=f$$ and call $N:C_0^\infty(\mathbb R^n)\to C^\infty(\mathbb R^n)$ the operator that sends $f$ to the special solution $w=\Gamma * f$, where $\Gamma$ is the fundamental solution. Consider also, for fixed indices $i,j$, the operator $$T_{ij}:C_0^\infty(\mathbb R^n)\to C^\infty(\mathbb R^n)$$ that sends $f$ to $D_{ij}Nf$. Then by the reference above, $T_{ij}$ is of strong type $(p,p)$, which means $\|T_{ij}f\|_p\leq C_p \|f\|_p$. In particular, by density of $C^\infty_0(\mathbb R^n)$ in $L^p(\mathbb R^n)$, $T_{ij}$ can be extended to $L^p(\mathbb R^n)$ to obtain that for any $f\in L^p$ $$\|D^2 w\|_p\leq C \|f\|_p$$ and you can conclude that also $Dw\in L^p$ e.g. by Gagliardo-Nirenberg. In this way you also have an estimate $\|w\|_{W^{2,p}}\leq C\|f\|_{L^p}$.
This is not exactly what you needed, but I hope it can be helpful anyway. For $\Omega$ bounded and Lipschitz I think you can use a partition of unity to reduce to the case in $\mathbb R^n$.