I spent a long morning at the library trying to understand why it so happens that the Cauchy problem for elliptic partial differential equations is not well posed. I still don't - I understand that it is, but what is the geometrical property of elliptic equations that makes it not well-posed, in contrast to parabolic or hyperbolic equations?
In the process I was not impressed by the creativity shown in textbooks. There are some good books out there - but the majority of them are isomorphic to each other. Most are well-written, a few stand out. But mostly they present the same topics in a different order, with more or less space devoted to each of them. Every preface should answer: "Why is this book different from all the other books?" Unfortunately, it seems that the question is not on most publishers' minds.
The good side is that there are plenty of open problems, even in areas that have already been studied a million times. Ask good questions, don't try to read all the literature, and have fun!