What is Solomon mind when he prayed the Lord to give his wisdom to judge ? Did he think the kingdom is His own ?
The account in the Bible you seem to refer to is 1 Kings chapter 3. The answer, as to whether king Solomon intended to have the baby cut in half, can be surmised from starting at verse 1, then considering the last verse in the same chapter.
The idea that he "pretended to cut the baby", has no basis, for there is no hint of any dramatic action when the sword Solomon requested was brought into the banquet hall. It was merely what he said to the sword-bearer that caused the genuine mother of the living baby to instantly speak up before any harm could come to her baby, proving the other woman's baby had died in the night, and she had been lying. Solomon merely had to speak, and the case was settled. That is what the account clearly shows.
As for the idea that he "had made up his mind to cut the baby", that is dispelled by the context of the chapter. Consider what God had granted to Solomon starting at verse 5. He appeared in a dream to Solomon to invite him to make a request of God. Solomon did not ask for wealth, or for long life, or for the life of his enemies, but for wisdom to judge the people of Israel. God was pleased to grant him that (and wealth with honour, besides). Just before the incident with the two women contesting the one living baby, God told Solomon:
"Lo, I have given to thee a heart, wise and understanding, that like thee there hath not been before thee, and after thee there doth not arise like thee" 1 Kings 3:11-12 Y.L.T.
The event of the baby and Solomon's judgment to ascertain the real mother is proof of God having granted that request of exceptional wisdom, in order to judge the people of God correctly. Consider the last verse of that chapter:
"And all Israel hear of the judgment that the king hath judged, and fear because of the king, for they have seen that the wisdom of God [is] in his heart, to do judgment." (Ibid, vs. 28)