Addition to my analysis below:

1. e7!!!
   ..Rg8, 2. Qf6+ Qg7, 3. e8=Q! Kh7 (3...Rxe8, 4. Qxg7# or 3....Qxf6, 4. Qxg8#), 4. Qh5+ Qh6, 5. Qhxh6#

Arein22, you are wrong because 3.e8=Q+ Kg7

1.e7 Rg8
2.e8=Q Rxe8
3.Qf6+ Qg7
4.Qxg7#

3.e8=Q#

2.Rxg8+ Kxg8(forced)

1.e7 Rg8

By the way this variaion is intresting:

e7 wins easily .

c) 1...Qxh4, 2. exf8=Q+ Kh7, 3. Qg7#

1. e7!!!

a) 1...Re8, 2. Qf6+ Qg7 (forced), 3. Qxg7#
b) 1...Rf7, 2. e8=Q+ Rf8, 3. Qxf8#