Consider a sequence of functions $f_n \in H^1(M)$, the first Sobolev space of a complete (possibly noncompact) Riemannian manifold. If we have the normalization $\Vert \nabla f_n\Vert_{L^2} = 1$, could we say that $\Vert f_n\Vert_{L^2}$ and $\Vert f_n\Vert_{L^p}$ are bounded, when $p \in (2, \frac{2n}{n - 2})$?
Edit: Could we say something even in the case $M = \mathbb{R}^n$ or $M = \mathbb{H}^n$? That would give me some idea as to how the proofs work.
The answer is not in general. For instance in $\mathbb{R}^d$ consider $f_n(x)=f(n^{-1}x)$ for some $f\in H^1$, then $\| f_n\|_p= n^{d/p}\| f\|_p$ and $\| \nabla f_n\|_2= n^{-1+d/2}\| \nabla f\|_2$. Therefore, if we define $g_n=n^{1-d/2}\| \nabla f\|_2^{-1}f_n$ then $\| \nabla g_n\|_2 =1$ and $\|g_n\|_p = n^{1+d/p-d/2} C$ which is clearly unbounded as long as $p<2d/(d-2)$.