Using the Calderon-Zygmund inequality, you can deduce $w\in W^{k,p}(\Omega)$ for $k$ large enough. Therefore you can deduce that $w\in L^{\frac{q}{2}}$  for some $q>n$ by Sobolev embedding. This skill can be seen in many elliptic pde's books, and your problem is not at the research level. You should put your problem in math stackexchange.