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

TL;DR

This paper investigates normalized solutions for a fractional Choquard equation with critical and supercritical nonlinearities, establishing nonexistence in the critical case and existence of ground states in the supercritical regime.

Contribution

It introduces new existence and nonexistence results for normalized solutions of fractional Choquard equations with critical and supercritical nonlinearities.

Findings

Nonexistence of solutions at the L^2-critical exponent.

Existence of normalized ground states in the supercritical range.

Development of variational methods for nonlocal fractional equations.

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.