scholarly journals Nonexistence Results for the Schrödinger-Poisson Equations with Spherical and Cylindrical Potentials in

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Yongsheng Jiang ◽  
Yanli Zhou ◽  
B. Wiwatanapataphee ◽  
Xiangyu Ge
Keyword(s):  
Poisson System ◽  
Nontrivial Solutions ◽  
Radial Functions ◽  
Poisson Equations

We study the following Schrödinger-Poisson system: , , , where are positive radial functions, , , and is allowed to take two different forms including and with . Two theorems for nonexistence of nontrivial solutions are established, giving two regions on the plane where the system has no nontrivial solutions.

Schrödinger–Poisson system with zero mass and convolution nonlinearity in R 2

2021 ◽  
pp. 1-21
Author(s):  
Heng Yang
Keyword(s):  
Ground State ◽  
Exponential Growth ◽  
Riesz Potential ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Ground State Solutions ◽  
Zero Mass ◽  
Subcritical Exponential Growth ◽  
New Ideas ◽  
Mass Case

In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass − Δ u + ϕ u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where α ∈ ( 0 , 2 ), I α : R 2 → R is the Riesz potential, f ∈ C ( R , R ) is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.

scholarly journals A neural network closure for the Euler-Poisson system based on kinetic simulations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Léo Bois ◽  
Emmanuel Franck ◽  
Laurent Navoret ◽  
Vincent Vigon
Keyword(s):  
Neural Network ◽  
Spatial Density ◽  
Data Generation ◽  
Poisson System ◽  
Mean Velocity ◽  
One Dimensional ◽  
Poisson Equations ◽  
Wide Range ◽  
The One ◽  
Uniform Accuracy

<p style='text-indent:20px;'>This work deals with the modeling of plasmas, which are ionized gases. Thanks to machine learning, we construct a closure for the one-dimensional Euler-Poisson system valid for a wide range of collisional regimes. This closure, based on a fully convolutional neural network called V-net, takes as input the whole spatial density, mean velocity and temperature and predicts as output the whole heat flux. It is learned from data coming from kinetic simulations of the Vlasov-Poisson equations. Data generation and preprocessings are designed to ensure an almost uniform accuracy over the chosen range of Knudsen numbers (which parametrize collisional regimes). Finally, several numerical tests are carried out to assess validity and flexibility of the whole pipeline.</p>

scholarly journals Existence of Solutions for a Schrödinger–Poisson System with Critical Nonlocal Term and General Nonlinearity

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Jiafeng Zhang ◽  
Wei Guo ◽  
Changmu Chu ◽  
Hongmin Suo
Keyword(s):  
Variational Method ◽  
Existence Of Solutions ◽  
Nonlocal Term ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Analysis Technique ◽  
General Nonlinearity

We study the existence and multiplicity of nontrivial solutions for a Schrödinger–Poisson system involving critical nonlocal term and general nonlinearity. Based on the variational method and analysis technique, we obtain the existence of two nontrivial solutions for this system.

scholarly journals Concentration results for a magnetic Schrödinger-Poisson system with critical growth

2020 ◽  
Vol 10 (1) ◽  
pp. 775-798
Author(s):  
Jingjing Liu ◽  
Chao Ji
Keyword(s):  
Variational Methods ◽  
Nonlinear Term ◽  
Type Equation ◽  
Critical Growth ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Penalization Technique ◽  
Poisson Type

Abstract This paper is concerned with the following nonlinear magnetic Schrödinger-Poisson type equation $$\begin{array}{} \displaystyle \left\{ \begin{aligned} &\Big(\frac{\epsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u=f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3,\\ &u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{array}$$ where ϵ > 0, V : ℝ3 → ℝ and A : ℝ3 → ℝ3 are continuous potentials, f : ℝ → ℝ is a subcritical nonlinear term and is only continuous. Under a local assumption on the potential V, we use variational methods, penalization technique and Ljusternick-Schnirelmann theory to prove multiplicity and concentration of nontrivial solutions for ϵ > 0 small.

Existence of nontrivial solutions for fractional Schrödinger-Poisson system with sign-changing potentials

2018 ◽  
Vol 41 (13) ◽  
pp. 5050-5064 ◽  
Author(s):  
Guofeng Che ◽  
Haibo Chen ◽  
Hongxia Shi ◽  
Zewei Wang
Keyword(s):  
Poisson System ◽  
Nontrivial Solutions

On the equations for non-linear wave solutions of the Vlasov and Poisson equations

1971 ◽  
Vol 6 (1) ◽  
pp. 119-124 ◽  
Author(s):  
R. J. Gribben
Keyword(s):  
Wave Speed ◽  
Linear Wave ◽  
System Of Equations ◽  
Poisson System ◽  
Wave Type ◽  
Poisson Equations ◽  
Wave Solutions ◽  
The Poisson Equation ◽  
Non Linear ◽  
Physical Importance

Butler & Gribben (1968) showed how to generalize non-linear, uniform, wave type solutions of the Vlasov–Poisson system of equations to allow for slow variations in the properties of the plasma. Here one equation of the derived set (that obtained by ensuring that the appropriate solutions of the Poisson equation are non-secular) is rearranged (i) so as to bring out the physical importance of those particles, trapped and untrapped, travelling near the wave speed, and (ii) so as to yield a form particularly suited to the case when the electric field is small.

Quasi-neutral limit of the non-isentropic Euler–Poisson system

2006 ◽  
Vol 136 (5) ◽  
pp. 1013-1026 ◽  
Author(s):  
Yue-Jun Peng ◽  
Ya-Guang Wang ◽  
Wen-An Yong
Keyword(s):  
Initial Data ◽  
Small Parameter ◽  
Time Interval ◽  
Cauchy Problems ◽  
Poisson System ◽  
Finite Time Interval ◽  
Smooth Solutions ◽  
Poisson Equations ◽  
Limit Problems ◽  
Formal Limit

This paper is concerned with multi-dimensional non-isentropic Euler–Poisson equations for plasmas or semiconductors. By using the method of formal asymptotic expansions, we analyse the quasi-neutral limit for Cauchy problems with prepared initial data. It is shown that the small-parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems have smooth solutions. Moreover, the formal limit is justified.

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