MULTIPLICITY OF NORMALIZED SOLUTIONS FOR THE FRACTIONAL SCHR?DINGER-POISSON SYSTEM WITH DOUBLY CRITICAL GROWTH

In this paper,we are concerned with solutions to the fractional Schr?dinger-Poisson system■ with prescribed mass ∫R3|u|2dx=a2,where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 <q <2s~*,and 2s~*=6/(3-2s) is the fractional critical Sobolev exponent.In the L2-subcritical case,we show the existence of multiple normalized solutions by using the genus theory and the truncation technique;in the L2-supercritical case,we obtain a couple of normalized solutions by developing a fiber map.Under both cases,to recover the loss of compactness of the energy functional caused by the doubly critical growth,we need to adopt the concentration-compactness principle.Our results complement and improve upon some existing studies on the fractional Schrodinger-Poisson system with a nonlocal critical term.