Assuming you intend instances of Application to be numbered, you could edit some or all of the theorem-related modules, adding a new theorem type named "Application" (along with a separate counter for it). The various theorem-related modules differ partly in how large a variety of theorem-like environments they provide (do they include Remark, Assumption, etc. or not?) and how numbering is done (globally, within sections, within environment types, ...). If you sometimes use one numbering system and sometimes another, and if you want Application defined in all systems you use, you will need to modify multiple modules.
On the other hand, if you just want a single heading available named Application (no numbering, no special formatting of what follows, just something analogous to Proof but without the end-of-proof mark), I would suggest creating a brand new module for it. That way, you could combine that module with any of the theorem-related modules. However you handle it, you will want to put the modified file(s) in the "layouts" directory under your personal LyX directory. Look in Help > About LyX for your "User directory" (typically ~/.lyx on Linux); there will be a "layouts" subdirectory under it. That will allow it to survive uninstall and reinstall episodes. For more details, refer to Help > Customization (chapter 5). Paul
