We can use find command to find the file and du -sh to find out its size.
We will execute du -sh on found files. So final command would be
find ~ -name "core.txt" -exec du -sh {} \;
or
find ~ -name "core.txt" | xargs du -sh
In 2nd command xargs will not handle spaces in file name. So We can tell exact delimiter to xargs to handle spaces in file name.
find ~ -name "core.txt" | xargs -d '\n' du -sh