For powers 2 ≤ p < 7/3, two sufficiently close radial positive energy minimizers of the generalized Choquard functional must be identical, and minimizing the energy is equivalent to saturating the Gagliardo-Nirenberg inequality.
Decay and Scattering in energy space for the solution of weakly coupled Schr\"odinger-Choquard and Hartree-Fock equations
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abstract
We prove decay with respect to some Lebesgue norms for a class of Schr\"odinger equations with non-local nonlinearities by showing new Morawetz inequalities and estimates. As a byproduct, we obtain large-data scattering in the energy space for the solutions to the systems of $N$ defocusing Schr\"odinger-Choquard equations with mass-energy intercritical nonlinearities in any space dimension and of defocusing Hartree-Fock equations, for any dimension $d\geq3$.
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Orbital stability of solitary waves for the generalized Choquard model
For powers 2 ≤ p < 7/3, two sufficiently close radial positive energy minimizers of the generalized Choquard functional must be identical, and minimizing the energy is equivalent to saturating the Gagliardo-Nirenberg inequality.