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