What is God's purpose in bringing the Jews back to Judah.

To restore them to the place God gave them after the exodus from Egypt. To fulfill one of the end times prophesies.

what is God's purpose in bringing the Jews back to Judah.

If you mean the restoration of the Jews is the promised land (Israel) in 1948, it was because God had promised to do so and made an everlasting covenant with the people of Israel.

I am not clear as to why you chose to limit it to one part of the land of Israel.

God has a general desire concerning ALL people:

1 Ti 2:2-5
3 This is good, and pleases God our Savior, 4 who wants all men to be saved and to come to a knowledge of the truth.
NIV

He did give some specific promises to Israel and Judah concerning possession of land which would be theirs as long as they obeyed the Law of Moses. AND/PLUS He gave the same promise to the Jews that He later gave to the Gentiles that if they trusted Him as their only God they would be saved.

Ro 9:27-32

27 Isaiah cries out concerning Israel:
"Though the number of the Israelites be like the sand by the sea,only the remnant will be saved . 28 For the Lord will carry outhis sentence on earth with speed and finality."
29 It is just as Isaiah said previously:
"Unless the Lord Almightyhad left us descendants,we would have become like Sodom,we would have been like Gomorrah."
30 What then shall we say? That the Gentiles, who did not pursue righteousness, have obtained it, a righteousness that is by faith; 31 but Israel, who pursued a law of righteousness, has not attained it. 32 Why not? Because they pursued it not by faith but as if it were by works. T
NIV

Why Judah alone is mentioned might be because of the dispersion of the so-called loss of the Ten Tribes (after the split of tribes) while Judah(and Benjamin) remained. But there are also promises that all of Israel's children would be called from all parts of the earth.

Why Judah alone is mentioned might be because of the dispersion of the so-called loss of the Ten Tribes (after the split of tribes) while Judah(and Benjamin) remained. But there are also promises that all of Israel's children would be called from all parts of the earth.

The ten tribes were not lost.