Albert Barnes Commentary Psalms 83:18

"That they may know that thou alone, whose name is Jehovah, Art the Most High over all the earth." — Psalms 83:18 (ASV)

That men may know - That all people may be impressed with the belief that you are the true and only God. This was the design and aim of the prayer in the psalm. It was that there might be such a manifestation of the power of God; that it might be so evident that the events which had occurred could be traced to no other source than God himself, that all people might be led to honor him.

That thou whose name alone is Jehovah - To whom alone this name belongs; to whom alone it can be properly ascribed. This was the special name by which God chose to be known (Exodus 6:3). Compare the notes at (Isaiah 42:8). On the word Jehovah - יהוה Yahweh - see the notes at (Psalms 68:4). It is found in combination, in (Genesis 22:14); (Exodus 17:15); (Judges 6:24); (Ezekiel 48:35); (Jeremiah 23:6); (Jeremiah 33:16).

Art the Most High over all the earth - You are the Supreme God, ruling over all people. Your dominion is so absolute over nations, even when combined together, and your power is so complete in foiling their plans, and disconcerting their purposes, that it is clear that you reign over them. He that could break up such a combination—he that could rescue his people from such an allied force—must have all power over the nations—must be the true God.