Studying differential equations without solving them: what, why, and how?

In this talk I will try to explain what it means to study PDEs (Partial Differential Equations) without actually solving them and why you might want to do it. I will concentrate on the particular type of PDEs which I mostly work with, called (degenerate) elliptic equations, covering some questions that people try to answer, and methods they (and myself in particular) use to do it. Finally, I will report on some recent progress I made during my sabbatical leave last year.