The functional equation P(f)=Q(g) in a p-adic field

Let K be a complete ultrametric algebraically closed field of characteristic π. Let P,Q be in Kx] with P′Q′ not identically 0. Consider two different functions f,g analytic or meromorphic inside a disk |x−a|<r (resp. in all K), satisfying P(f)=Q(g). By applying the Nevanlinna's values distribution Theory in characteristic π, we give sufficient conditions on the zeros of P′,Q′ to assure that both f,g are “bounded” in the disk (resp. are constant). If π≠2 and deg(P)=4, we examine the particular case when Q=λP (λ∈K) and we derive several sets of conditions characterizing the existence of two distinct functions f,g meromorphic in K such that P(f)=λP(g).