Test: open browser, http://localhost:8080/server-smoke/, push the button 'Please don't smoke'
Admin console: http://localhost:8080
TomEE is a small and fast server with decent Java EE 7 support (however it is not formally certified). TomEE is based on Tomcat and comes with comparable footprint.
The fully loaded "plus" edition is only ~17 MB larger (download and disk size), than the stripped down webprofile download.
Commercial support is available for TomEE directly from tomitribe -- the source.