I have been able to find these two but I don't know how valuable they are as a reference, "Problems and examples in differential equations" By Biler, and, "Partial Differential Equations through Examples and Exercises" by Pap. However, I don't know if these are any good, or the ones that I'm looking for, maybe you guys can answer that. Side note: these is my second course on PDEs, my first course covered the usual topics in a standard PDE class for undergraduates, and was less rigorous than what's expected as a graduate student.
I also found unpublished handbooks, with problems and their solutions, that are occasionally encountered in graduate level courses in ODE, and PDE, by UCLA Prof. Yanovsky, that deals with more theoretical questions on PDEs that I am looking for. So a textbook along the lines of these notes should suffice.
I'm hoping there is more textbooks, or more lecture notes, out there which focuses more on the theoretical side of PDE. Similar to what these General Topology supplemental texts provided, "Elementary Topology Problem Textbook" by Viro, and, "Fundamentals of General Topology: Problems and Exercises" by Arkhangel'skii. I doubt any theory focused PDE book will contain a solutions manual. I am focused on a textbook that is at the level of "Partial Differential Equations" by Fritz.