Normalized solutions for a class of fractional Choquard equations with the HLS lower critical term and a nonlocal perturbation

Abstract

In this paper, we study the mass-constrained fractional Choquard equation \( (-\Delta)^s u = \lambda u + \alpha (I_\mu * |u|^{\frac{2N-\mu}{N}})|u|^{\frac{2N-\mu}{N}-2}u + (I_\mu * |u|^p)|u|^{p-2}u \) in \( \mathbb{R}^N \), under the constraint \( \int_{\mathbb{R}^N} |u|^2 \, dx = c^2 > 0 \), where \( N > 2s \), \( s \in (0,1) \), \( \mu \in (0,N) \), \( \alpha > 0 \), and \( 2 + \frac{2s-\mu}{N} \le p < \frac{2N-\mu}{N-2s} \). We first establish a nonexistence result in the \( L^2 \)-critical case \( p = 2 + \frac{2s-\mu}{N} \). Then, in the \( L^2 \)-supercritical range, we prove the existence of normalized ground states in two complementary regimes determined by the quantity \( \mathcal{M}_1(c) \). Our approach is based on constrained variational methods, a min-max construction, and refined estimates for the associated fiber maps.