For a bounded smooth domain $\Omega$, let $H\_2^{s}(\Omega)$ be the usual Sobolev space on $\Omega$.


Define $A:=\{f\in H\_2^{s}(\Omega)| \lVert f\rVert\_{L^p(\Omega)}=1\}$ where $2<p<2\_{s}^\*$.


Can we show $A$ is a $C^2$-Hilbert manifold since I need to use the Morse lemma?



Is an $L^p$-sphere in Sobolev space $H_2^{s}(\Omega)$ a Hilbert manifold?
The norm map
$$
f\mapsto \int\_{\Omega} f^p dvol
$$
is a $C^k$-map for $k<p$ and in fact smooth if $p$ is an even integer as is shown in [this paper](http://www.jstor.org/stable/24901438.) of Bonic and Frampton. The implicit function theorem for maps of Banach spaces then applies to give your desired result.


