Large solutions for biharmonic maps in four dimensions

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