There exist constructive and non-constructive proofs.
Sometimes, for a mathematical statement, we can have both non-constructive and a constructive proof.
However, are there statements for which there is only a non-constructive proof? (The fact that there maybe a construction of the required object but we don't know it yet doesn't count here).
Phrased differently, are there statements (that claim existence of objects) that are essentially non-constructive?