Is it possible to run a web application using Sun ONE application server 7 in Intellij Idea Community Edition 10.5? I want to configure the server in IDE and run the application there.
No. For running/debugging/deploying to application servers, you'll need the Ultimate Edition.
Having said that, with Community Edition you can configure an external tool "File > Settings > Tools > External Tools", which is basically a shortcut from the IDE to your application server and runs it in a console in the IDE.