A Sharp Mass Threshold for Fractional Choquard Equations with Lower Hardy-Littlewood-Sobolev and \(L^2\)-Critical Terms

Abstract: For (\frac12\le s<1), we study a fractional Choquard equation with prescribed mass and nonlinearities at the lower Hardy-Littlewood-Sobolev and (L^2)-critical exponents. The sharp Hardy-Littlewood-Sobolev and Choquard Gagliardo-Nirenberg inequalities determine an explicit critical mass (a_). For (0<a\le a_), we compute the exact infimum of the constrained energy and prove that it is not attained and that no normalized solution exists. For (a>a_*), the energy is unbounded from below on the mass sphere, while the Pohozaev set is nonempty.