Normalized Solutions and Semiclassical Concentration for Upper-Critical Fractional Choquard Equations

Abstract

We study a fractional Choquard equation with an upper-critical Hartree term, an $L^2$-supercritical Hartree perturbation, and a semiclassical potential under a prescribed $L^2$-mass constraint. The potential is bounded and nonnegative, has a nonempty zero set, and has a positive lower limit at infinity. For every prescribed mass and all sufficiently small semiclassical parameters, we prove the existence of a pair of normalized solutions $\pm u_{\epsilon }$ with a negative Lagrange multiplier. The proof combines a strict energy bound below the critical one-bubble level, compactness modulo translations for the autonomous ground-state set, a simultaneous cutoff of both Hartree terms, and a localized constrained mountain-pass argument. Moreover, suitable translates of $u_{\epsilon }$ converge strongly in $H^s({\mathbb{R}}^N)$ to a positive autonomous ground state, and the corresponding concentration points approach the zero set of the potential as $\epsilon \to 0$.

24 pages, comments are welcome