Normalized Solutions and Critical Growth in Fractional Nonlinear Schrödinger Equations with Potential

We investigate the existence of positive normalized (mass-constrained) solutions for the fractional nonlinear Schrödinger equation (−Δ)sv+V(x)v=λv+μ|v|p−2v+|v|2s*−2vinRN,∥v∥22=b2, where N>2s, s∈(0,1), μ>0, p∈(2,2s*), and 2s*=2NN−2s. Here, λ∈R denotes the Lagrange multiplier associated with the prescribed mass b>0. The potential V∈C1(RN) is allowed to be nonconstant and satisfies V(x)→V∞ as |x|→∞; moreover, the perturbations induced by V−V∞ and x·∇V are assumed to be small in the quadratic-form sense compared with the fractional Dirichlet form ∥(−Δ)s/2v∥22. Using the Caffarelli–Silvestre extension, we establish a Pohozaev identity adapted to the presence of V(x) and introduce a Pohozaev manifold on the L2-sphere. Combining Jeanjean’s augmented functional approach with a splitting analysis at the Sobolev-critical level, we construct compact Palais–Smale sequences below a suitable critical energy level. As a consequence, we prove the existence of positive normalized solutions for small masses b∈(0,b0) in the L2-critical and L2-supercritical regimes (with respect to the lower-order power p).