8 Remember that Jesus Christ of the seed of David was raised from the dead according to my gospel:
9 Wherein I suffer trouble, as an evil doer, even unto bonds; but the word of God is not bound.
10 Therefore I endure all things for the elect's sakes, that they may also obtain the salvation which is in Christ Jesus with eternal glory.
Paul suffered trouble as an evil doer from his brethren according to the flesh, the Jews. Paul endured all this suffering that even the Jews, God's elect of the seed of David, might also obtain the salvation which is in Christ Jesus. The elect in this context is the nation of Israel. God elected Israel, through the seed of David, to be the people of flesh to birth the Messiah, Jesus Christ. Elect simply means God's choosing. Paul longed for the Jews to be saved.
Romans 9
2 That I have great heaviness and continual sorrow in my heart.
3 For I could wish that myself were accursed from Christ for my brethren, my kinsmen according to the flesh:
4 Who are Israelites; to whom pertaineth the adoption, and the glory, and the covenants, and the giving of the law, and the service of God, and the promises;
5 Whose are the fathers, and of whom as concerning the flesh Christ came, who is over all, God blessed for ever. Amen.