<p>I need to use JavaScript in a form for mobile web application.</p>

<p>I need some information about it, related to browser compatibility as well as way to use JavaScript in mobile web applications (syntax).</p>

It's a nightmare. It's like working with web-browsers in the 90's but with the manager expectations of tomorrow.
I strongly recommend you determine your target platforms in terms of regions, software, and actual hardware since the market is much more fragmented than the web and a large number of mobile devices have issues with CSS, JS and even XHTML ranging from painful to not-on-this-platform.
The good news is that android and iphone are pushing the field rapidly. The bad news is that makes a lot of the sources you'll google badly out-of-date.
Some references:
http://www.w3.org/TR/mobileOK-basic10-tests/#test_objects_or_script
http://mobiforge.com/developing/story/getting-started-with-mobile-ajax
http://mobiforge.com/
Especially that last one :)