The function L[ϕ,m0,m1,m2] is defined in terms of P02,P11
and P20
, not of P1,P2 and P3.
Assuming that P02 = P1; P11 = P2; P20 = P3
you can do it this way:
Clear[\[Phi], m0, m1, m2, L, DL];
L[\[Phi]_, m0_, m1_, m2_] =
P1[\[Phi]]^m0*P2[\[Phi]]^m1*P3[\[Phi]]^m2;
DL[\[Phi]_, m0_, m1_, m2_] =
Simplify[D[L[\[Phi], m0, m1, m2], \[Phi]], Element[\[Phi], Reals]];
Reduce[DL[\[Phi], m0, m1, m2] == 0 &&
Element[{m0, m1, m2}, PositiveIntegers] &&
Element[\[Phi], Reals],
\[Phi]]