Evans Pde Solution Chapter 6 Zip 1 2 !EXCLUSIVE!
Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations may be an excelent option for you, since you look for theory connected to funtional analysis. The last three chapters are about PDE's, using functional analysis tools developed in the beggining of the book. It has a huge amount of beutiful exercises, with most of them solved in the end of the book. May be just what you are looking for.
evans pde solution chapter 6 zip 1 2
This is more of a lengthy comment than an answer, because I fear you won't get a satisfactory one. Solved problems in PDE-books are a kind of rare beast. Apart from some simple calculations in the beginning of a course, in general the problems are theoretical in nature and at this point in your studies, you should be able to judge the correctness of your proof alone. (At least this would be my attempt at an explanation. The real answer is probably that books on PDE normally are quite lengthy anyway, as are the solutions to many exercises, and neither author nor publisher want to write or respectively pay for even more pages...)
Brezis has already been mentioned, apart from this, of all the PDE-books I know, there are none with solutions. The only thing which also comes to mind is "Linear functional analysis" by Alt, which is a recent translation of a German classic. I have only read the original, but it has solved exercises and it touches some topics also related to PDE.
However I would say that those kind of calculations are a bit outdated, since they usually only solve simple cases and do neither help with understanding the general concepts nor really help in practical applications. Modern study of PDE instead mostly consists of proving existence with an abstract argument and then maybe showing some further properties of the solution, for example regularity. There is also the numerical analysis side of things, however while some of the methods are the same (After all, if you want your numerical algorithm to converge to a solution, there needs to exist one in the first place), I have always experienced them as a different kind of crowd. Usually they are interested in solving the same few classical problems over and over again, but with more and more improved algorithms. And I am not sure, if you would count the source code of some numerical solver as solution to a given problem.
For more advanced problems around the level of 1st or 2nd year graduate studies, looking up solutions to end of the chapter problems in Evans "PDEs" text is quite easy. There are many many resources online for finding answers, more so than any book I am familiar with.