Elliptic regularity with negative Sobolev space on bounded or unbounded domains

Question

I am looking for some reference which deals with the existence and regularity of solution to $ -\Delta u = f $ in bounded or unbounded domain $\Omega$ and with Dirichlet boundary condition, $u|\partial \Omega = 0$ where $f \in W^{-1,p}(\Omega)$ (or some other negative sobolev space in general)

Answer 1

I highly recommend François Treves' and Hörmander's books on these topics, but the ones you need are:

• Treves, F. (1975). Basic Linear Partial Differential Equations. Academic Press. In Chapter 3, it discusses this exact problem, although only in the case $f \in W^{-1, 2}(\Omega)$, where $\Omega$ is bounded.
• Agranovich, M. S. (2015). Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains. Springer. This text delves into Dirichlet problems for elliptic operators in general, but also only considers the case $f \in W^{-1, 2}(\Omega)$, where $\Omega$ is a bounded Lipschitz domain.
• Simader, C. G., & Sohr, H. (1996). The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains. CRC Press. This book discusses a slew of related problems, but there is no mention of such $W^{-1, p}$ spaces.

I also skimmed many other articles and books, and as far as I can tell, there is no systematic investigation of the case where $f \in W^{-1, p}(\Omega)$, beyond the special case where $p = 2$ and $\Omega$ is a bounded Lipschitz domain.