Exited States to Fractional NLS Equations with a Prescribed L2-Norm and a Degenerate External Potential

We investigate the existence and local uniqueness of normalized $k$-peak solutions for the fractional Schr\"odinger equations with attractive interactions with a class of degenerated trapping potential with non-isolated critical points. Precisely, we first construct the $k$-peak concentrated solutions without any constraint through the finite dimensional reduction method. Then, various local Pohozaev identities are established, which are used to give the limit relationship between the parameters and prove the existence of the normalized solutions to the original problem. Finally, we prove the local uniqueness of the $k$-peak solutions with prescribed $L^2$-norm. The nonlocal properties of fractional order operators are fully exploited in this work. We optimize the range of fractional order index, and distinguish the different cases of $p-1\frac{4s}N$, which are called respectively that the mass-subcritical, the mass-critical, and the mass-supercritical case.