The qualitative behavior for \alpha-harmonic maps from a surface with boundary into a sphere

Let u_\alpha be a sequence of smooth \alpha-harmonic maps from a compact Riemann surface M with boundary \partial M to a compact Riemannian manifold N with free boundary u_\alpha (\partial M) on a supporting submanifold \Gamma of N and with uniformly bounded \alpha-energy. If the target manifold N is a sphere S^{K-1}, we show that there is no energy loss for such a sequence of maps during the blow-up process as \alpha \searrow 1. Moreover, the image of the weak limit map and bubbles is a connect set. Also, the case of Dirichlet boundary is considered.