Min-max n-harmonic maps of degree 1 with free-boundary into mathbb{S}^(n-1) in almost round balls

Let $n\geq 3$ and let $\Omega \subset \mathbb{R}^n$ be a $\mathcal{C}^1$ bounded domain which is diffeomorphic to a ball. We investigate here the problem of finding critical points of the $n$-energy in the space $\mathcal{I}=\{v\in W^{1,n}(\Omega,\mathbb{R}^n) ; \ |\mathrm{tr}_{|\partial \Omega}v|=1\}$. Maps in $\mathcal{I}$ have a well-defined topological degree on $\partial \Omega$ but this degree is not continuous for the weak convergence in $W^{1,n}$. Hence finding critical points with prescribed degrees results in a problem of lack of compactness. We first prove that minimizers of the $n$-energy exist only when $\Omega$ is a round ball and when the prescribed degree is $-1,0$ or $1$. We then develop a mountain pass approach for the $(n+\alpha)$-energies and study the convergence, when $\alpha$ goes to zero, of the resulting critical points via a bubbling analysis. We exclude the existence of bubbles in the case where $\Omega$ is close to a ball by proving an energy gap result for free boundary $n$-harmonic maps from $\mathbb{B}^n$ to $\mathbb{B}^n$. We thus obtain the existence of critical points of the $n$-energy with prescribed degree $1$ when $\Omega$ is close to a ball.