Blow-up profile of normalized solutions for fractional nonlinear Schrödinger equation with negative potentials

We investigate the existence and blow-up profile of normalized solutions to the fractional nonlinear Schrödinger equationalign cases (-Δ)ˢu+V(x)u+λ u = |u|4s/Nu, \ in\ RN, \\ ∫RN|u|²dx = α, cases alignwith N≥ 2, s∈(0, 1), α>0, λ, and negative potentials $ V(x) $. Firstly, we prove the existence and nonexistence of normalized solutions for negative potentials V∈ Lᵖ(RN)+Lq(RN) with N/2s≤ p<q < ∞ in the mass-critical setting. Secondly, we refine the nonexistence results and analyze the asymptotic behavior of the minimizers α\ as α α^* for two types of potentials: a bounded potential V(x) = -η(1+|x|²)ˢ, and a singular potential V(x) = -γ₁|x-a₁|-m₁-γ₂|x-a₂|-m₂. Our study provides precise energy estimates and sharp blow-up rates in the convergence of minimizers α\.