The East View database provides searchable full text of many Russian social science publications.  The search interface is not especially sophisticated but a general search using "Карамазовы" will bring up over 600 matches.  This is a database of Russian resources so you will be limited to Russian language materials here. 

 The Hathi Trust is searched as a part of the general search from the Main Library's search interface.  However, if you search this interface in Russian you will get an inaccurate return on the search.  Nevertheless, you can search it in transliteration and get a pretty accurate view of what is available via the HathiTrust. 

It is important to note that full text is not available for every title here.  To see those works with full text available you should refine your search to include just those items under "Full View". 

While this source has limitations it can be very valuable if you are seeking early editions of literary works.  For example, here you see a general search on the work "Карамозовы".  As you can see it returns over 600 matches, one of which is the original publication of the novel in the journal Русский Вестник.

It is also possible to create "collections" of related materials for easy retrieval later.  The Hathi Trust draws is not limited to the materials digitized by Google Books.  The partner institutions have contributed journals and other texts that may not appear via Google.

JSTOR (Journal Storage) is one of th most important digital depositories (subscription based) for UIUC researchers.  In addition to journals, this digital library also contains books and primary sources.  As of April 16, 2015, JSTOR provides full-text access to approximately 20 Slavic Studies journals and 6680 journals for Language & Literature.  Access: http://www.library.illinois.edu/sfx?r=63750000000000343.