Large solutions for biharmonic maps in four dimensions

Abstract

We seek critical points of the Hessian energy functional \(E_\Omega(u)\!=\!\int_\Omega\vert\Delta u\vert^2dx\), where \(\Omega={\mathbb R}^4\) or Ω is the unit disk \(B\) in \({\mathbb R}^4\) and u : Ω → S 4. We show that \(E_{{\mathbb R}^4}\) has a critical point which is not homotopic to the constant map. Moreover, we prove that, for certain prescribed boundary data on ∂B, E B achieves its infimum in at least two distinct homotopy classes of maps from B into S 4.