To prove the smoothness of an $\alpha$-harmonic map, Sachs and Uhlenbeck firstly show (in their paper "The existence of minimal immersions of 2-spheres") that it is in the Sobolev space $L^2_2(M,N)$, then show that it is also in $L^4_2(M,N)$. Finally, they use 'elliptic bootstrapping' to show that it is smooth.
My question is why it is in $L^4_2(M,N)$, $M$ being a compact surface. To show this, they start from the Euler-Lagrange equation satisfied by the $\alpha$-harmonic map $s$ and consider the 'linear operator' $\Delta_s \colon L^4_2(M,N) \to L^4_0(M,N)$, where $$\Delta_su=\Delta u + (\alpha-1)(\langle d^2u,ds \rangle)ds/(1+|ds|^2)$$
They mention that since $\Delta_s$ has a nice inverse, one has $s \in L^4_2(M,N)$, but give no details. Can someone give a detailed proof of this statement? J. D. Moore gives an outline of the proof in his book "Introduction to Global Analysis - minimal surfaces in Riemannian manifolds", pp.214-5. He essentially says $A(s)(ds,ds)$ is in $L^4_0$, $A$ being the second fundamential form of $N$ in some Euclidean space, and prove that $\Delta_s$ is invertible and hence surjective, so $s$ is in $L^4_2(M,N)$.
I do not understand the proof, due to the following:
(1) $\Delta_s$ is not a 'linear operator', since $L^4_2(M,N)$ is a manifold.
(2) It seems $\Delta_s$ is not injective, since the operator maps all constant maps (to $N$) to $0$.
(3) $A(s)(ds,ds)$ may not be in $L^4_0(M,N)$, because the former is normal to $T_{s(x)}N$, $x \in M$. So, it may be some other $L^4_0$ spaces instead.
