I doubt there's a closed form for the equation for general constant a and P.
a = 65/100; P = 1;
Solve[{(a ProductLog[R])^(1/(1 - a)) + (a ProductLog[R])^(a/(1 - a)) == P}, R, Reals]
(* {{R -> E^Root[{-13 + #1^7 &, -5 + #2^7 &, -2 + #3^7 &,
800 #1 #2^6 #3^5 - 3380 #4^13 - 2197 #4^20 &}, {1, 1, 1,
2}]^7 Root[{-13 + #1^7 &, -5 + #2^7 &, -2 + #3^7 &,
800 #1 #2^6 #3^5 - 3380 #4^13 - 2197 #4^20 &}, {1, 1, 1, 2}]^7}} *)
%//N
{{R -> 3.57197}}
