arXiv subjects
Publications and source records attributed to George Venkov.
We prove the decay in the energy space for the solution to the defocusing biharmonic Hartree-Fock equations with mass-supercritical and energy-subcritical Choquard-type nonlinearity in space dimension $d\geq3$. We treat both the free and the perturbed by a potential case. As a direct consequence, we obtain large-data scattering in $H^2(\mathbb R^d)^N, \, N\geq 1$.
We consider the generalized Choquard equation describing trapped electron gas in 3 dimensional case. The study of orbital stability of the energy minimizers (known as ground states) depends essentially in the local uniqueness of these minimizers. In equivalent way one can optimize the Gagliardo--Nirenberg inequality subject to the constraint fixing the $L^2$ norm. The uniqueness of the minimizers for the case $p=2$, i.e. for the case of Hartree--Choquard is well known. The main difficulty for the case $p > 2$ is connected with possible lack of control on the $L^p$ norm of the minimizers.
We prove decay with respect to some Lebesgue norms for a class of Schrödinger 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 Schrödinger-Choquard equations with mass-energy intercritical nonlinearities in any space dimension and of defocusing Hartree-Fock equations, for any dimension $d\geq3$.
We study the $p$-Choquard equation in 3-dimensional case and establish existence and uniqueness of ground states for the corresponding Weinstein functional. For proving the uniqueness of ground states, we use the radial symmetry to transform the equation into an ordinary differential system, and applying the Pohozaev identities and Gronwall lemma we show that any two Weinstein minimizers coincide.
In the present article we study the radial symmetry of minimizers of the energy functional, corresponding to the repulsive Hartree equation in external Coulomb potential. To overcome the difficulties, resulting from the "bad" sign of the nonlocal term, we modify the reflection method and then, by using Pohozaev integral identities we get the symmetry result.