Minimizing fibrations and $p$-harmonic maps in homotopy classes from $S^3$ into $S^2$

We prove that, contrary to the case of maps from 5 into S', there exist infinitely many homotopy classes from S into S having a minimizing 3-harmonic map. We prove that the first eigenforms of the linear operator A/ = d* on (Ker d) (1 A5"* are stable for the associated conformal invariant non-linear variational problem and we deduce, in particular, that the Hopf map from 5 into 5 minimizes the p-energy in it's homotopy class for p ^ 4 and that it remains true locally for 3 ^ p < 4. We prove that the Hopf map minimizes the p-energy for p ^ 3 among a class of symmetric fibrations from S into S.