Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential

We are concerned with the following nonlinear fractional Schrödinger equation: <tex-math id="FE1"> {document}equation (-Δ)ˢ u+V(x)u+ω u = |u|ᵖ⁻²u in\,\,RN,\;\;\;\;\;\;(P)equation {document} </tex-math> where <tex-math id="M1">{document}s∈(0,1){document}</tex-math> and <tex-math id="M2">{document}p∈(2+4s/N,2^*ₛ){document}</tex-math>, that is, the mass supercritical and Sobolev subcritical. Under certain assumptions on the potential <tex-math id="M3">{document}V:RN→ R{document}</tex-math>, positive and vanishing at infinity including potentials with singularities (which is important for physical reasons), we prove that there exists at least one <tex-math id="M4">{document}L²{document}</tex-math>-normalized solution <tex-math id="M5">{document}(u,ω)∈ Hˢ(RN)×R⁺{document}</tex-math> 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.