Excerpts from Books and Wikipedia
"After securing his internal borders and incorporating outlying northern regions into his kingdom, David set about making international conquests. Family feuds resulted in David choosing his son, Solomon, by Bathsheba as his heir after the rebellion by his eldest son, Absalom." [Historical Atlas of Judaism, p. 74]
"Saul had no intention of facing down this giant, but another Israelite accepted the challenge: David, the younger brother of three siblings from Judah who had joined Saul’s army. . . . This victory made David so popular that Saul decided to get rid of him, as a possible competitor for the throne. David, to preserve his own life, fled into Philistine territory." [Bauer: Ancient World, p. 319-21]