  This article investigates the Fourier extension operator associated to the
fractional surface $(\xi,|\xi|^{\alpha})$ with $\alpha\geq 2$. We show that
nearly all valid scale-invariant Fourier extension inequalities possess
extremals. More precisely, if the Fourier extension operator is bounded for an
endpoint $p_0$, then for all $1<p< p_0$, the corresponding $L^p$-extremal
sequences are precompact up to symmetries. In particular, we have developed the
relevant $L^p$-profile decomposition for these Fourier extension operators.
