188 research outputs found
Groundstates of the Choquard equations with a sign-changing self-interaction potential
We consider a nonlinear Choquard equation when
the self-interaction potential is unbounded from below. Under some
assumptions on and on , covering and being the one- or
two-dimensional Newton kernel, we prove the existence of a nontrivial
groundstate solution by solving a
relaxed problem by a constrained minimization and then proving the convergence
of the relaxed solutions to a groundstate of the original equation.Comment: 16 page
The semirelativistic Choquard equation with a local nonlinear term
We propose an existence result for the semirelativistic Choquard equation
with a local nonlinearity in \begin{equation*} \sqrt{\strut
-\Delta + m^2} u - mu + V(x)u = \left( \int_{\mathbb{R}^N}
\frac{|u(y)|^p}{|x-y|^{N-\alpha}} \, dy \right) |u|^{p-2}u - \Gamma (x)
|u|^{q-2}u, \end{equation*} where and the potential is decomposed
as the sum of a -periodic term and of a bounded term that decays
at infinity. The result is proved by variational methods applied to an
auxiliary problem in the half-space
Light bullets in the spatiotemporal nonlinear Schrodinger equation with a variable negative diffraction coefficient
We report approximate analytical solutions to the (3+1)-dimensional spatiotemporal nonlinear Schr\"odinger equation, with the uniform self-focusing nonlinearity and a variable negative radial diffraction coefficient, in the form of three-dimensional solitons. The model may be realized in artificial optical media, such as left-handed materials and photonic crystals, with the anomalous sign of the group-velocity dispersion (GVD). The same setting may be realized through the interplay of the self-defocusing nonlinearity, normal GVD, and positive variable diffraction. The Hartree approximation is utilized to achieve a suitable separation of variables in the model. Then, an inverse procedure is introduced, with the aim to select a suitable profile of the modulated diffraction coefficient supporting desirable soliton solutions (such as dromions, single- and multilayer rings, and multisoliton clusters). The validity of the analytical approximation and stability of the solutions is tested by means of direct simulations
Semirelativistic Choquard equations with singular potentials and general nonlinearities arising from Hartree-Fock theory
We are interested in the general Choquard equation \begin{multline*}
\sqrt{\strut -\Delta + m^2} \ u - mu + V(x)u - \frac{\mu}{|x|} u
=
\left( \int_{\mathbb{R}^N} \frac{F(y,u(y))}{|x-y|^{N-\alpha}} \, dy \right)
f(x,u) - K (x) |u|^{q-2}u \end{multline*} under suitable assumptions on the
bounded potential and on the nonlinearity . Our analysis extends
recent results by the second and third author on the problem with and
pure-power nonlinearity . We show that, under appropriate
assumptions on the potential, whether the ground state does exist or not.
Finally, we study the asymptotic behaviour of ground states as
A guide to the Choquard equation
We survey old and recent results dealing with the existence and properties of
solutions to the Choquard type equations and some of its variants and extensions.Comment: 39 page
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