where each separate letter represents a different digit and A is not zero.

What is ABCDE?

I did a search for all 5 digit numbers which are equal to the sum of the fith powers of its digits.  And only found these

54748 92727 and 93084

of these 3 only 93084 has unique digits so the answer is

A=9 B=3 C=0 D=8 and E=4