On the nonlocal Schrödinger‐poisson type system in the Heisenberg group

On the nonlocal Schr\”{o}dinger-Poisson type system in the Heisenberg group

10.22541/au.162062954.40403164/v1 ◽

2021 ◽

Author(s):

Zeyi Liu ◽

Min Zhao ◽

Deli Zhang ◽

Sihua Liang

Keyword(s):

Heisenberg Group ◽

Bounded Domain ◽

Type System ◽

Smooth Bounded Domain ◽

Kohn Laplacian ◽

Poisson Type

This paper is concerned with the following nonlocal Schr\”{o}dinger-Poisson type system: \begin{equation*} \begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}dx\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u, &\mbox{in} \ \Omega,\\ -\Delta_{H}\phi=u^2 & \mbox{in}\ \Omega,\\ u=\phi=0 & \mbox{on}\ \partial\Omega, \end{cases} \end{equation*} where $a, b>0$ and $\Delta_H$ is the Kohn-Laplacian on the first Heisenberg group $\mathbb{H}^1$, $\Omega\subset \mathbb{H}^1$ is a smooth bounded domain, $\lambda>0$, $\mu\in \mathbb{R}$ are some real parameters and $1“”

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The Schrödinger-Poisson type system involving a critical nonlinearity on the first Heisenberg group

Israel Journal of Mathematics ◽

10.1007/s11856-020-1961-8 ◽

2020 ◽

Vol 235 (1) ◽

pp. 385-411

Author(s):

Yu-Cheng An ◽

Hairong Liu

Keyword(s):

Heisenberg Group ◽

Type System ◽

Critical Nonlinearity ◽

Poisson Type

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Sign-changing solutions for a Schrödinger–Kirchhoff–Poisson system with 4-sublinear growth nonlinearity

Electronic journal of qualitative theory of differential equations ◽

10.14232/ejqtde.2021.1.86 ◽

2021 ◽

pp. 1-21

Author(s):

Shubin Yu ◽

Ziheng Zhang ◽

Rong Yuan

Keyword(s):

Type System ◽

Invariant Sets ◽

Smooth Domain ◽

Positive Parameter ◽

Poisson System ◽

Bounded Smooth Domain ◽

Existence And Multiplicity ◽

Bounded Smooth ◽

Sign Changing Solutions ◽

Poisson Type

In this paper we consider the following Schrödinger–Kirchhoff–Poisson-type system { − ( a + b ∫ Ω | ∇ u | 2 d x ) Δ u + λ ϕ u = Q ( x ) | u | p − 2 u in Ω , − Δ ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω , where Ω is a bounded smooth domain of R 3 , a > 0 , b ≥ 0 are constants and λ is a positive parameter. Under suitable conditions on Q ( x ) and combining the method of invariant sets of descending flow, we establish the existence and multiplicity of sign-changing solutions to this problem for the case that 2 < p < 4 as λ sufficient small. Furthermore, for λ = 1 and the above assumptions on Q ( x ) , we obtain the same conclusions with 2 < p < 12 5 .

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Multiplicity and concentration results for a class of critical fractional Schrödinger–Poisson systems via penalization method

Communications in Contemporary Mathematics ◽

10.1142/s0219199718500785 ◽

2018 ◽

Vol 22 (01) ◽

pp. 1850078 ◽

Cited By ~ 1

Author(s):

Vincenzo Ambrosio

Keyword(s):

Continuous Function ◽

Positive Solutions ◽

Fractional Laplacian ◽

Type System ◽

Laplacian Operator ◽

Positive Potential ◽

Penalization Method ◽

Subcritical Growth ◽

Fractional Laplacian Operator ◽

Poisson Type

We deal with the multiplicity and concentration of positive solutions for the following fractional Schrödinger–Poisson-type system with critical growth: [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text], [Formula: see text], [Formula: see text], with [Formula: see text], is the fractional Laplacian operator, [Formula: see text] is a continuous positive potential and [Formula: see text] is a superlinear continuous function with subcritical growth. Using penalization techniques and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions with the topology of the set where the potential attains its minimum value.

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Ultrastructural characteristics of sporidia of Ustilago

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100155906 ◽

1989 ◽

Vol 47 ◽

pp. 786-787

Author(s):

Patricia N. Hackney

Keyword(s):

Electron Microscope ◽

Type System ◽

Tube Formation ◽

High Voltage Electron Microscopy ◽

Voltage Electron ◽

University Of Wisconsin ◽

Transmission Electron ◽

Ustilago Hordei ◽

Silene Alba ◽

The University

Ustilago hordei and Ustilago violacea are yeast-like basidiomycete pathogens ofHordeum vulgare and Silene alba respectively. The mating type system in both species of Ustilago is bipolar, with alleles, A,a, (U.hordei) and a1, a2 (U.violacea) at a single locus. Haploid sporidia maintain the asexual phase by budding, while the sexual phase is initiated by conjugation tube formation between the mating types during budding and conjugation.For observation of budding, sporidia were prepared by culturing the four types on YEG (yeast extract glucose) broth for 24 hours. After centrifugation at 5000g cells were either left unmated or mated in a1/a2,A/a combinations. The sporidia were then mixed 1:1 with 4% agar and the resulting 1mm cubes fixed in 8% gluteraldehyde and post fixed in osmium tetroxide. After dehydration and embedding cubes were thin sectioned with a LKB ultratome and photographed in a Zeiss 9s transmission electron microscope or in an AE1 electron microscope of MK11 1MEV at the High Voltage Electron Microscopy Center of the University of Wisconsin-Madison.

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