There's no simple solution - but a work-around
Google site search is a simple way to add a search functionality to your website. Site search through Adsense also gives you a part of the income from advertising (as shown in the search results, see "Adsense for Search" or " "). However, Site search depends on the normal Google results to function. This means that if your site is not fully (or recently) indexed in Google, the site search results will be incomplete (or completely empty) - leaving only advertising for competing websites to be shown.
It is advisable to work on getting your site fully indexed in Google in any case, however this is often a long-term target. A possible solution for the time until then is to use a non-search engine site-search system like "Express Search" from WebSideStory.