Poisson System Research Articles

Multiplicity and concentration of nontrivial solutions for Kirchhoff–Schrödinger–Poisson system with steep potential well

Multiplicity and concentration of nontrivial solutions for Kirchhoff–Schrödinger–Poisson system with steep potential well

A fast point charge interacting with the screened Vlasov–Poisson system

A fast point charge interacting with the screened Vlasov–Poisson system

Groundstates of a magnetic critical Choquard Poisson system with multiple potentials

Groundstates of a magnetic critical Choquard Poisson system with multiple potentials

Existence and asymptotic behavior of solutions for nonhomogeneous Schrödinger–Poisson system with exponential and logarithmic nonlinearities

Existence and asymptotic behavior of solutions for nonhomogeneous Schrödinger–Poisson system with exponential and logarithmic nonlinearities

Publisher's Note: “A new method for solving the linearized 1D Vlasov–Poisson system yielding a new class of solutions” [Phys. Plasmas 31, 052121 (2024)]

Publisher's Note: “A new method for solving the linearized 1D Vlasov–Poisson system yielding a new class of solutions” [Phys. Plasmas <b>31</b>, 052121 (2024)]

Approximate solutions for the Vlasov–Poisson system with boundary layers

We construct the approximate solutions to the Vlasov–Poisson system in a half-space, which arises in the study of the quasi-neutral limit problem in the presence of a sharp boundary layer, referred as to the plasma sheath in the context of plasma physics. The quasi-neutrality is an important characteristic of plasmas and its scale is characterized by a small parameter, called the Debye length. We present the approximate equations obtained by a formal expansion in the parameter and study the properties of the approximate solutions. Moreover, we present numerical experiments demonstrating that the approximate solutions converge to those of the Vlasov–Poisson system as the parameter goes to zero.

Construction of Ground-State Solutions of the Gross–Pitaevskii–Poisson System Using Genetic Algorithms

Construction of Ground-State Solutions of the Gross–Pitaevskii–Poisson System Using Genetic Algorithms

Existence and stability of weak solutions of the Vlasov–Poisson system in localised Yudovich spaces

We consider the Vlasov–Poisson system both in the repulsive (electrostatic potential) and in the attractive (gravitational potential) cases. Our first main theorem yields the analog for the Vlasov–Poisson system of Yudovich’s celebrated well-posedness theorem for the Euler equations: we prove the uniqueness and the quantitative stability of Lagrangian solutions f=f(t,x,v) whose associated spatial density ρf=ρf(t,x) is potentially unbounded but belongs to suitable uniformly-localised Yudovich spaces. This requirement imposes a condition of slow growth on the function p↦‖ρf(t,⋅)‖Lp uniformly in time. Previous works by Loeper, Miot and Holding–Miot have addressed the cases of bounded spatial density, i.e. ‖ρf(t,⋅)‖Lp≲1 , and spatial density such that ‖ρf(t,⋅)‖Lp∼p1/α for α∈[1,+∞) . Our approach is Lagrangian and relies on an explicit estimate of the modulus of continuity of the electric field and on a second-order Osgood lemma. It also allows for iterated-logarithmic perturbations of the linear growth condition. In our second main theorem, we complement the aforementioned result by constructing solutions whose spatial density sharply satisfies such iterated-logarithmic growth. Our approach relies on real-variable techniques and extends the strategy developed for the Euler equations by the first and fourth-named authors. It also allows for the treatment of more general equations that share the same structure as the Vlasov–Poisson system. Notably, the uniqueness result and the stability estimates hold for both the classical and the relativistic Vlasov–Poisson systems.

Normalized Solutions for Schrödinger–Poisson Systems Involving Critical Sobolev Exponents

Normalized Solutions for Schrödinger–Poisson Systems Involving Critical Sobolev Exponents

Infinitely many sign-changing solutions for planar Schrödinger–Poisson system

In this paper, we consider the following planar Schrödinger–Poisson system { − Δ u + V ( x ) u + ϕu = | u | q − 2 u , in R 2 , Δϕ = u 2 , in R 2 , where the potential V : R 2 → R is continuous and coercive. By using the method of perturbation and invariant sets of descending flow, we obtain the existence of infinitely many sign-changing solutions for this system.

Infinitely Many Solutions for Schrödinger–Poisson Systems and Schrödinger–Kirchhoff Equations

By applying Clark’s theorem as altered by Liu and Wang and the truncation method, we obtain a sequence of solutions for a Schrödinger–Poisson system −Δu+V(x)u+ϕu=f(u)inR3,−Δϕ=u2inR3 with negative energy. A similar result is also obtained for the Schrödinger-Kirchhoff equation as follows:−1+∫RN∇u2Δu+V(x)u=f(u)u∈H1(RN).

Saddle solutions for the planar Schrödinger–Poisson system with exponential growth

Saddle solutions for the planar Schrödinger–Poisson system with exponential growth

ZK‐type limit for the Euler–Poisson system with a realistic electrostatic potential in dimension three

We explore the Zakharov–Kuznetsov‐type limit for the three‐dimensional Euler–Poisson systems with external magnetic fields when the plasma has a realistic electrostatic potential. It reveals that both homogeneous and inhomogeneous Zakharov–Kuznetsov‐type equations can be obtained from the Euler–Poisson system through the application of the Gardner–Morikawa transformation. By analyzing the remaining system, we employ the energy method within Sobolev spaces to provide estimates, leading to the emergence of Zakharov–Kuznetsov‐type dynamics over a time interval of the order .

Schrödinger–Poisson systems with zero mass in the Sobolev limiting case

AbstractWe study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schrödinger equation driven by a weighted ‐Laplace operator and without the mass term, and a higher‐order fractional Poisson equation. Since the system is set in , the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a logarithm, hence sign‐changing and unbounded from above and below. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation, where the logarithm is uniformly approximated by polynomial kernels. Our results are new even in the planar case .

Multiple bound states for a class of fractional critical Schrödinger–Poisson systems with critical frequency

In this paper we study the fractional Schrödinger–Poisson system ε2s(−Δ)su+V(x)u=ϕ|u|2s*−3u+|u|2s*−2u,ε2s(−Δ)sϕ=|u|2s*−1,x∈R3, where s ∈ (0, 1), ɛ &amp;gt; 0 is a small parameter, 2s*=63−2s is the critical Sobolev exponent and V∈L32s(R3) is a nonnegative function which may be zero in some regions of R3, e.g., it is of the critical frequency case. By virtue of a new global compactness lemma, and the Lusternik–Schnirelmann category theory, we relate the number of bound state solutions with the topology of the zero set where V attains its minimum for small values of ɛ.

Existence and Asymptotic Behavior of Solutions for a Quasilinear Schrödinger–Poisson System in ℝ3 with a General Nonlinearity

Existence and Asymptotic Behavior of Solutions for a Quasilinear Schrödinger–Poisson System in ℝ3 with a General Nonlinearity

Smooth Subsonic and Transonic Spiral Flows with Nonzero Vorticity to Steady Euler–Poisson System in Concentric Cylinders

Smooth Subsonic and Transonic Spiral Flows with Nonzero Vorticity to Steady Euler–Poisson System in Concentric Cylinders

Global-in-time error estimates of non-relativistic limits for Euler–Maxwell system near non-constant equilibrium

It was proved that Euler–Maxwell systems converge globally-in-time to Euler–Poisson systems near non-constant equilibrium states when the speed of light c→∞. In this paper, we establish the global-in-time error estimates between smooth solutions of Euler–Maxwell systems and those of Euler–Poisson systems near non-constant equilibrium states. The main difficulty lies in the singularity of the error variable for the electric field E, so that more careful estimates for the time derivatives of error variables should be established. The proof takes good advantage of the anti-symmetric structure of the error system and an induction argument on the order of the derivatives.

Non-radial ground state solutions for fractional Schrödinger–Poisson systems in $$\mathbb {R}^{2}$$

Non-radial ground state solutions for fractional Schrödinger–Poisson systems in $$\mathbb {R}^{2}$$

Normalized solutions for a fractional Schrödinger–Poisson system with critical growth

In this paper, we study the fractional critical Schrödinger–Poisson system (-Δ)su+λϕu=αu+μ|u|q-2u+|u|2s∗-2u,inR3,(-Δ)tϕ=u2,inR3,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}{\\left\\{ \\begin{array}{ll} (-\\Delta )^su +\\lambda \\phi u= \\alpha u+\\mu |u|^{q-2}u+|u|^{2^*_s-2}u,&{}~~ \\hbox {in}~{\\mathbb {R}}^3,\\\\ (-\\Delta )^t\\phi =u^2,&{}~~ \\hbox {in}~{\\mathbb {R}}^3,\\end{array}\\right. } \\end{aligned}$$\\end{document}having prescribed mass ∫R3|u|2dx=a2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _{{\\mathbb {R}}^3} |u|^2dx=a^2,\\end{aligned}$$\\end{document}where s,t∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ s, t \\in (0, 1)$$\\end{document} satisfy 2s+2t>3,q∈(2,2s∗),a>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\,s+2t> 3, q\\in (2,2^*_s), a>0$$\\end{document} and λ,μ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda ,\\mu >0$$\\end{document} parameters and α∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in {\\mathbb {R}}$$\\end{document} is an undetermined parameter. For this problem, under the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-subcritical perturbation μ|u|q-2u,q∈(2,2+4s3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu |u|^{q-2}u, q\\in (2,2+\\frac{4\\,s}{3})$$\\end{document}, we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. In the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-supercritical perturbation μ|u|q-2u,q∈(2+4s3,2s∗)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu |u|^{q-2}u,q\\in (2+\\frac{4\\,s}{3}, 2^*_s)$$\\end{document}, we prove two different results of normalized solutions when parameters λ,μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda ,\\mu $$\\end{document} satisfy different assumptions, by applying the constrained variational methods and the mountain pass theorem. Our results extend and improve some previous ones of Zhang et al. (Adv Nonlinear Stud 16:15–30, 2016); and of Teng (J Differ Equ 261:3061–3106, 2016), since we are concerned with normalized solutions.