What was leonidas’s objective in the Persian war

To hold up the Persian land advance, and force the Persian navy to try to turn the position by sea. The Greek naval forces lay in wait in the strait of Artemesion nearby to engage the Persian fleet when it moved.

Destruction of the Persian fleet was vital as it threatened the Greek cities, and they kept their armies at home to protect themsels individually from sea attack, and so could not be persuaded to concentrate against the Persian army.

In fact the sea battle at Artemesion failed and the Greek fleet retired to reform and try again at Salamis. There being no further need to hold the pass at Thermopylai, Leonidas sent the Greek land forces home and stayed behind with his Spartan and Thespian contingents to cover this withdrawal. They were eliminated, but the rest got away safely.