If I interpret "same page" to mean "page that looks like this one" rather than "current page without even reloading", the the answer is pretty simple; in fact the documentation for the CGI module shows pretty much this approach.

#!/usr/bin/perl -wT
use strict;
use CGI qw/:standard/;
$|++;

print header, start_html;             

###
### Generate your page - everything up to the <div> you want - here
###

if (my $search = param('search')){
    ###
    ### Retrieve the results based on the search term
    ###
    print div("search_results go here");
}

###
### The content you want after the search <div> goes here
###

print end_html;
[download]

Otherwise, ikegami and the others are right - it's a JS question.

-- 
Human history becomes more and more a race between education and catastrophe. -- HG Wells