On \(p\)-harmonic self-maps of spheres (Q6039201)

\(p\)-harmonic maps are critical points of the energy \[ E_{p}(\phi) = \frac{1}{p} \int_M |d\phi|^p \] and act as a natural generalisation of harmonic maps, which correspond to \(p=2\). This is corroborated by the conformal invariance of \(p\)-harmonic maps on \(p\)-dimensional domains. But, for \(p\neq 2\), the Euler-Lagrange equation degenerates at critical points and the regularity of \(p\)-harmonic maps is more complicated. The existence question of \(p\)-harmonic maps remains a largely open question and this article takes on the case of maps from the standard Euclidean sphere to itself. Parametrizing the sphere by spherical coordinates naturally leads to a particular class of self-maps which only vary the colatitude. In this set-up, the problem of finding \(p\)-harmonic maps becomes a second-order ODE with specific boundary values. While the equation is arduous, the main features of its non-linear terms allow the use of a Lyapunov function. Then all hinges on a control of the oscillation of the associated pendulum to prove the existence of an infinite number of \(p\)-harmonic maps from \({\mathbb S}^m\) to itself when \(p < m < 2 + p + 2\sqrt{p}\). Beyond these values, such symmetric solutions do not exist. A precise description of the equivariant stability of these \(p\)-harmonic maps is also given.