Advanced book on partial differential equations

Here are some suggestions.

1. Functional Analysis, Sobolev Spaces, and Partial Differential Equations by Haim Brezis. This violates your rule of not developing the functional analysis material, but is a very good book. You can skip the stuff you know and jump right to the PDE / operator bits.
2. An Introduction to Partial Differential Equations by Michael Renardy and Robert Rogers. Here you want the last part of the book, say after chapter 8. There's a lot of nice stuff in Chapters 10-12 that uses lots of functional analysis to solve nonlinear elliptic problems, etc.
3. Monotone Operators in Banach Space and Nonlinear PDE by Ralph Showalter. This is heavy functional / operator theoretic material used to solve some serious nonlinear problems.
4. Nonlinear Differential Equations of Monotone Type in Banach Spaces by Biorel Barbu. This covers the same sort of material as the Showalter book.
5. Applications of Functional Analysis and Operator Theory by Hutson and Pym. There's a lot more in here than applications in PDE, but you might find it interesting.