You didn't clarify what agent you were using to "search" for your documents. There are no wildcard searches supported by Finder, as none are supported by Spotlight, upon which the Finder depends for searching. The Terminal application does support wildcard searches owing to the Shell and UNIX operating system.
Luis Sequeira1 has covered the Finder's Find (⌘F) interface as one approach.
When using the Finder Window search field, or the Spotlight menu bar 🔍 tool, one can incorporate a case-insensitive search with a partial name to match a file pattern in a given folder:
(kind:folder name:downloads) OR (kind:pdf name:profile)
Apple has some documentation on using Spotlight searches: Narrow your search results on Mac - Apple Support
If done in a Finder Window, that will display a resulting window with several Downloads folders and also the PDF files found in the folder matching your partial name parameter. When you have the Finder View menu > Show Path Bar enabled, clicking on a resulting file will show the path to it at the bottom of the window.