On a class of mixed Choquard-Schrödinger-Poisson systems

@article{Ghergu2019OnAC,
  title={On a class of mixed Choquard-Schr{\"o}dinger-Poisson systems},
  author={Marius Ghergu and Gurpreet Singh},
  journal={Discrete \& Continuous Dynamical Systems - S},
  year={2019}
}
We study the system $$ \left\{ -\Delta u+u+K(x) \phi |u|^{q-2}u&=(I_\alpha*|u|^p)|u|^{p-2}u &&\mbox{ in }{\mathbb R}^N, -\Delta \phi&=K(x)|u|^q&&\mbox{ in }{\mathbb R}^N, \right. $$ where $N\geq 3$, $\alpha\in (0,N)$, $p,q>1$ and $K\geq 0$. Using a Pohozaev type identity we first derive conditions in terms of $p,q,N,\alpha$ and $K$ for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach. 
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References

SHOWING 1-10 OF 33 REFERENCES
Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency
We study the nonlocal Schrödinger–Poisson–Slater type equation $$\begin{aligned} - \Delta u + (I_\alpha *\vert u\vert ^p)\vert u\vert ^{p - 2} u= \vert u\vert ^{q-2}u\quad \text {in}\quad \mathbb…
Semi-classical states for the Choquard equation
TLDR
If the external potential V has a local minimum and p is a small parameter then for all small varepsilon > 0$$ε>0 the problem has a family of solutions concentrating to the local minimum of V.
Ground state solutions for non-autonomous fractional Choquard equations
We consider the following nonlinear fractional Choquard equation, \begin{equation}\label{e:introduction} \begin{cases} (-\Delta)^{s} u + u = (1 + a(x))(I_\alpha \ast (|u|^{p}))|u|^{p - 2}u\quad\text{…
Pointwise Bounds and Blow-up for Choquard-Pekar Inequalities at an Isolated Singularity
Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity
Abstract We study the following nonlinear Choquard equation: - Δ ⁢ u + V ⁢ ( x ) ⁢ u = ( 1 | x | μ ∗ F ⁢ ( u ) ) ⁢ f ⁢ ( u )   in ⁢ ℝ N , $-\Delta u+V(x)u=\biggl{(}\frac{1}{|x|^{\mu}}\ast…
Classification of Positive Solitary Solutions of the Nonlinear Choquard Equation
AbstractIn this paper, we settle the longstanding open problem concerning the classification of all positive solutions to the nonlinear stationary Choquard equation $$\Delta…
Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
Positive solutions for some non-autonomous Schrödinger–Poisson systems
Strongly interacting bumps for the Schrödinger–Newton equations
We study concentrated bound states of the Schrodinger–Newton equations h2Δψ−E(x)ψ+Uψ=0, ψ>0, x∊R3; ΔU+12|ψ|2=0, x∊R3; ψ(x)→0, U(x)→0 as |x|→∞. Moroz et al. [“An analytical approach to the…
Effective One Particle Quantum Dynamics of Electrons: A Numerical Study of the Schrodinger-Poisson-X alpha Model
The Schrödinger-Poisson-Xα (S-P-Xα) model is a “local one particle approximation” of the time dependent Hartree-Fock equations. It describes the time evolution of electrons in a quantum model…
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