Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential

<p style='text-indent:20px;'>We are concerned with the following nonlinear fractional Schrödinger equation:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{equation} (-\Delta)^s u+V(x)u+\omega u = |u|^{p-2}u\quad {\rm{in}}\,\,{\mathbb{R}}^N,\;\;\;\;\;\;({\textbf{P}})\end{equation}$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ s\in(0,1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ p\in\left(2+4s/N,2^*_s\right) $\end{document}</tex-math></inline-formula>, that is, the mass supercritical and Sobolev subcritical. Under certain assumptions on the potential <inline-formula><tex-math id="M3">\begin{document}$ V:{\mathbb{R}}^N\rightarrow {\mathbb{R}} $\end{document}</tex-math></inline-formula>, positive and vanishing at infinity including potentials with singularities (which is important for physical reasons), we prove that there exists at least one <inline-formula><tex-math id="M4">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-normalized solution <inline-formula><tex-math id="M5">\begin{document}$ (u,\omega)\in H^s({\mathbb{R}}^N)\times{\mathbb{R}}^+ $\end{document}</tex-math></inline-formula> of equation (P). In order to overcome the lack of compactness, the proof is based on a new min-max argument and splitting lemma for nonlocal version.</p>