Existence of normalized solutions for large masses

Establish the existence of normalized solutions for the fractional Choquard equation with lower Hardy–Littlewood–Sobolev exponent q_L=(N+α)/N and L²-critical exponent t_c=(N+α+2s)/N, subject to the mass constraint \(\|u\|_2^2=a\), for every mass \(a>a_*\), where \(a_*\) is the sharp critical mass threshold.

Background

The paper studies a fractional Choquard equation containing two nonlocal nonlinearities: a lower Hardy–Littlewood–Sobolev critical term and an L²-critical Choquard term. The authors determine a sharp mass threshold $a_*$. For masses $0<a\le a_*$, they compute the constrained energy infimum, prove that it is not attained, and establish nonexistence of normalized solutions. For $a>a_*$, they prove that the energy is unbounded below on the mass sphere and that the Pohozaev set is nonempty.

Because the constrained energy is unbounded below for $a>a_*$, direct minimization cannot yield solutions. Although every normalized solution must belong to the Pohozaev set, the authors note that nonemptiness of this set does not establish that it is a natural constraint for the energy. Consequently, existence of normalized solutions for all masses above the threshold remains unresolved.