Does  Drupal have an AJAX solution for search results, preferably with Apachesolr integration?
Depending on your needs, there are a couple of solutions.  Probably the closest thing to Apple's Live Search feature is the Finder module.  It does not integrate with Apache Solr at this time, however.

I would also keep your eye on the Inline Ajax Search module as Solr integration is on the roadmap.  It might need some contributions to push integration through.

A third solution to try is the Apache Solr AJAX module.  It might not fit exactly what you are trying to do, however it allows for inline refreshing of the page when queries are entered and filters are enabled.