Normalized ground states for the lower critical fractional Choquard equation with a focusing local perturbation

In this paper, we study the existence of normalized ground states to following lower critical fractional Choquard equation $ (-\Delta)^su = \lambda u+\gamma(I_{\alpha}*|u|^{1+\frac{\alpha}{N}})|u|^{\frac{\alpha}{N}-1}u+\mu |u|^{q-2}u\ \mbox{in}\ \mathbb R^N under L^2 $-norm constraint$ \int_{\mathbb R^N}|u|^2dx a^2, $where N \geq3 $, s\in(0,1) \alpha\in (0,N) a, \gamma, \mu>0 and 2<q\leq 2_s^*: 2N/(N-2s) $. Under suitable restrictions on \gamma \mu prove nonexistence, symmetry states. Specifically, using extremal function with construction technique, establish radially without any $-subcritical perturbation, i.e. 2<q<2+4s/N $-supercritical case 2+4s/N<q<2_s^* introduce homotopy-stable family Palais-Smale sequence, compactness sequence illustrate particular, consider Sobolev q 2_s^* which corresponds equations involving double terms is rarely studied in existing literatures. With aid subcritical approximation method, also obtain