See Biblica: Robert J. Merecz, «Jezebel’s Oath (1 Kgs 19,2)«Jezebel’s Oath (1 Kgs 19,2)», Vol. 90 (2009) 257-259 for a detailed discussion of this subject.
After careful analysis of the text, Merecz concludes that Jezebel's goal was not to kill Elijah, but to frighten him into a hasty self-imposed exile. By running away, Elijah would leave his followers confused and undermine his victory at Mt. Carmel. Rather than sending an assassin, she cleverly sends him a dire-sounding warning which has the desired effect. Even her intimidating oath is subtly phrased so that she is not under the oath herself.
