Jesse R. Elrod's military information is located at the National Archives and Records Administration, in a microfilm of "Compiled Service Records of Confederate Soldiers Who Served in Organizations From the State of Arkansas."
You can get a copy of his Compiled Service Record (CSR) by clicking on the big, white banner at the top of this page -- this is the cheapest and fastest way. You can also order it direct from the National Archives, but it's pricey and takes a long time.
There's no record of an Arkansas Confederate pension application from the widow of Jesse R. Elrod. If she moved to another Southern state after the war, she may have applied there.