A weak energy identity for $(n+\alpha )$-harmonic maps with a free boundary in a sphere

In this article, we show that sequences of $(n+\alpha )$-harmonic maps with a free boundary in ${\mathbb{S}}^{d-1}$, where $\alpha$ is a parameter tending to zero, converge to a bubble tree. For such sequences, we prove in detail that the limiting energy is equal to the energy of the macroscopic limit plus the sum of the energies of certain ``bubbles’', each multiplied by a corresponding coefficient.