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 ${(-\Delta )}^{s}v+V\left(x\right)v=\lambda v+{\mu \left|v\right|}^{p-2}v+{\left|v\right|}^{2_s^{*}-2}v\mathrm{in}{\mathbb{R}}^N,{\parallel v\parallel }_2^2=b^2,$ where $N>2s$, $s\in (0,1)$, $\mu >0$, $p\in (2,2_s^{*})$, and $2_s^{*}={\displaystyle \frac{2N}{N-2s}}$. Here, $\lambda \in \mathbb{R}$ denotes the Lagrange multiplier associated with the prescribed mass $b>0$. The potential $V\in C^1\left({\mathbb{R}}^N\right)$ is allowed to be nonconstant and satisfies $V\left(x\right)\to V_{\infty }$ as $\left|x\right|\to \infty$; moreover, the perturbations induced by $V-V_{\infty }$ and $x·\nabla V$ are assumed to be small in the quadratic-form sense compared with the fractional Dirichlet form ${\parallel (-\Delta )}^{s/2}{v\parallel }_2^2$. Using the Caffarelli–Silvestre extension, we establish a Pohozaev identity adapted to the presence of $V\left(x\right)$ and introduce a Pohozaev manifold on the $L^2$-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 $b\in (0,b_0)$ in the $L^2$-critical and $L^2$-supercritical regimes (with respect to the lower-order power p).