Ground‐state solution of a nonlinear fractional Schrödinger–Poisson system

2021 ◽  
Author(s):  
Haidong Liu ◽  
Leiga Zhao
Keyword(s):  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Poisson System

Existence and Concentration of Positive Ground State Solutions for Schrödinger-Poisson Systems

2013 ◽  
Vol 13 (3) ◽  
Author(s):  
Jun Wang ◽  
Lixin Tian ◽  
Junxiang Xu ◽  
Fubao Zhang
Keyword(s):  
Small Parameter ◽  
Ground State ◽  
Ground State Solution ◽  
Real Parameter ◽  
State Solution ◽  
Poisson System ◽  
Ground State Solutions ◽  
Subcritical Nonlinearity

AbstractIn this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger-Poisson systemwhere ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that b(x) has a maximum. We prove that the system has a positive ground state solution

scholarly journals Multiple solutions for a quasilinear Schrödinger–Poisson system

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Zhang
Keyword(s):  
Nontrivial Solution ◽  
Ground State ◽  
Multiple Solutions ◽  
Continuous Functions ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Poisson System ◽  
Subcritical Growth ◽  
Monotonicity Properties

AbstractIn this article, we consider the following quasilinear Schrödinger–Poisson system $$ \textstyle\begin{cases} -\Delta u+V(x)u-u\Delta (u^{2})+K(x)\phi (x)u=g(x,u), \quad x\in \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2}, \quad x\in \mathbb{R}^{3}, \end{cases} $$ { − Δ u + V ( x ) u − u Δ ( u 2 ) + K ( x ) ϕ ( x ) u = g ( x , u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , where $V,K:\mathbb{R}^{3}\rightarrow \mathbb{R}$ V , K : R 3 → R and $g:\mathbb{R}^{3}\times \mathbb{R}\rightarrow \mathbb{R}$ g : R 3 × R → R are continuous functions; g is of subcritical growth and has some monotonicity properties. The purpose of this paper is to find the ground state solution of (0.1), i.e., a nontrivial solution with the least possible energy by taking advantage of the generalized Nehari manifold approach, which was proposed by Szulkin and Weth. Furthermore, infinitely many geometrically distinct solutions are gained while g is odd in u.

scholarly journals Existence of ground state solutions for an asymptotically 2-linear fractional Schrödinger–Poisson system

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dandan Yang ◽  
Chuanzhi Bai
Keyword(s):  
Continuous Function ◽  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Poisson System ◽  
Ground State Solutions ◽  
Linear Condition

AbstractIn this paper, we investigate the following fractional Schrödinger–Poisson system: $$\left \{ \textstyle\begin{array}{l@{\quad}l} (-\Delta)^{s} u + u + \phi u = f(u), & \text{in } \mathbb{R}^{3}, \\ (-\Delta)^{t} \phi= u^{2}, & \text{in } \mathbb{R}^{3}, \end{array}\displaystyle \right . $${(−Δ)su+u+ϕu=f(u),in R3,(−Δ)tϕ=u2,in R3, where $\frac{3}{4} < s < 1$34<s<1, $\frac{1}{2} < t < 1$12<t<1, and f is a continuous function, which is superlinear at zero, with $f(\tau) \tau \ge3 F(\tau) \ge0$f(τ)τ≥3F(τ)≥0, $F(\tau) = \int_{0}^{\tau} f(s) \,ds$F(τ)=∫0τf(s)ds, $\tau \in\mathbb{R}$τ∈R. We prove that the system admits a ground state solution under the asymptotically 2-linear condition. The result here extends the existing study.

scholarly journals Existence of Positive Ground State Solution for the Nonlinear Schrödinger-Poisson System with Potentials

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Lu Xiao ◽  
Haiqin Qian ◽  
Mengmeng Qu ◽  
Jun Wang
Keyword(s):  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Poisson System ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger

In the present paper we study the existence of positive ground state solutions for the nonautonomous Schrödinger-Poisson system with competing potentials. Under some assumptions for the potentials we prove the existence of positive ground state solution.

Exact Ground State Solution for Anyons with spin-$\frac {1}{2}$

1992 ◽  
Vol 17 (1) ◽  
pp. 107-110
Author(s):  
Shun-Qing Shen ◽  
J. Cai ◽  
Rui-Bao Tao
Keyword(s):  
Ground State ◽  
Ground State Solution ◽  
State Solution ◽  
Exact Ground State
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