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.