The very weak solution like
-\Delta u=f in \Omega, u=0 on \partial\Omega,
when f \in L^p with 1\le p
About Moser iteration technique,
After you build several formal grammar rules (e.g., context-free grammar for compiler) for semantics and syntax inteactive processes, you always need prove the grammar is regular expression (that is not easy, but you can refer to the techniques of formal language), and then give some testing functions for the grammar to find the next states and output after iterative process.
Why do not you use standard Calderon-Zygmund estimates to get W^{2,p} regularity with p>1? As for Moser iteration for any f\in W^{-1,q} with q>1 try xi=G(u)={u+k, u
Give a look at this paper
http://www.ams.org/mathscinet/search/publdoc.html?arg3=1992&co4=AND&co5=AND&co6=AND&co7=AND&dr=pubyear&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=di%20fazio&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq
and let me know if you get trouble.
Best,
Giuseppe
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