ApacheSolr Search results with AJAX (D7)
Apache Solr AJAX module is available for D6

Project page says


  The Apache Solr AJAX module AJAXifies Apache Solr Search Integration –
  if you change your query or filters, you see the new result set
  instantly – no need to reload the full page.


For D7 there exists a Sandbox Project 

You can try the module, Download module using below commands, 

git clone --branch master http://git.drupal.org/sandbox/darthsteven/1116776.git apache_solr_ajax___d7
cd apache_solr_ajax___d7


Found an article here Drupal7 AJAXified Solr Search with facets. Search API, Panels and some custom code. you can give a try..