I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts:
Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean space.
1) For $p > 1$ the space $W^{1,p}(M,N)$ is a Banach manifold.
2) For $p>m$ the space $W^{1,p}(M,N)$ is a $C^\infty$ Banach manifold.
3) For $p>m$ the $p$-energy functional $\int_M|du(x)|^pd\mu$ satisfies the Palais-Smale condition.
The reference that she gives, Palais' Foundations of Global Nonlinear Analysis is too daunting a read at this time for me. There seem to be no other references where these facts are proved. If anyone could give a proof of any of these three ( especially 2) or 3) ), it would be much appreciated. A reference other than Palais' would also be much appreciated.