Infinite families of harmonic self-maps of spheres

For each of the spheres Sn, n≥5, we construct a new infinite family of harmonic self-maps, and prove that their members have Brouwer degree ±1 or ±3. These self-maps are obtained by solving a singular boundary value problem. As an application we show that for each of the special orthogonal groups SO(4), SO(5), SO(6) and SO(7) there exist two infinite families of harmonic self-maps.