If $u \in H_0^1(\Omega) \cap L^p(\Omega)$ and $\Delta u \in L^p(\Omega)$ then $u \in W^{2, p}(\Omega)$

Question

How to show the following?

If $u \in H_0^1(\Omega) \cap L^p(\Omega)$, $\Delta u \in L^p(\Omega)$ then $u \in W^{2, p}(\Omega)$.

This is part of the Brezis-Kato regularity argument as presented by Struwe in Variational Methods, Appendix B, but I am unable to prove it.

Any hints will be the most appreciated.