Nonexistence, existence and symmetry of normalized ground states to Choquard equations with a local perturbation

We study the Choquard equation with a local perturbation −Δu=λu+(Iα∗|u|p)|u|p−2u+μ|u|q−2u,x∈RN having prescribed mass ∫RN|u|2dx=a2. For L2-critical or L2-supercritical μ|u|q−2u, we prove nonexistence, existence and symmetry of normalized ground states, by using mountain pass lemma, Pohožaev constraint method, Schwartz symmetrization rearrangements some theories polarizations. In particular, our results cover Hardy-Littlewood-Sobolev upper critical exponent case p=(N+α)/(N−2) for N≥3. Our are nonlocal counterpart in Li [Studies solutions to Schrödinger equations Sobolev combined nonlinearities. 2021 Apr 28, arXiv:2104.12997v2]; Soave [Normalized states NLS nonlinearities: case. J Funct Anal. 2020;279(6):Article 108610]; Wei Wu mixed 2021. arXiv:2102.04030v1].