Radially symmetric σ2,p-harmonic maps from n-dimensional annuli into sphere

Abstract

Consider a bounded Lipschitz domain $A^n\subset R^n$ and the ${\sigma }_{2,p}$-energy functional $F_{{\sigma }_{2,p}}[u;A^n]≔{\int }_{A^n}|\nabla u\wedge \nabla {u|}^{p}dx,$with $p\in ]1,\infty ]$, defined over the space of admissible Sobolev maps $A_p\left(A^n\right)≔\left\{u\in W^{1,2p}(A^n,S^{n-1}):u{|}_{\partial A^n}=\frac{x}{\left|x\right|}\right\}.$In this paper, we investigate the multiplicity and uniqueness of extremals and strong local minimisers of the ${\sigma }_{2,p}$-energy functional $F_{{\sigma }_{2,p}}[\cdot ,A^n]$ in $A_p\left(A^n\right)$. Our focus is on the space of admissible Sobolev maps and a topological class of maps known as spherical twists in connection with the Euler–Lagrange equations associated with the ${\sigma }_{2,p}$-energy functional over $A_p\left(A^n\right)$, referred to as ${\sigma }_{2,p}$-harmonic map equation on $A^n$. Our main result reveals a surprising difference between even and odd dimensions, showing infinitely many smooth solutions in even dimensions and only one in odd dimensions. This result is based on a careful analysis of the full versus restricted Euler–Lagrange equations.