Nontrivial solutions for the fractional Schrödinger‐Poisson system with subcritical or critical nonlinearities

2019 ◽  
Vol 43 (3) ◽  
pp. 1484-1494
Author(s):  
Zi‐an Fan
Keyword(s):  
Poisson System ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities

2020 ◽  
Vol 23 (3) ◽  
pp. 723-752 ◽  
Author(s):  
Alessio Fiscella ◽  
Patrizia Pucci
Keyword(s):  
Scalar Case ◽  
Fractional Operator ◽  
Nontrivial Solutions ◽  
Fractional Systems ◽  
Lack Of Compactness ◽  
Critical Nonlinearities ◽  
Kirchhoff Model

AbstractThis paper deals with the existence of nontrivial solutions for critical possibly degenerate Kirchhoff fractional (p, q) systems. For clarity, the results are first presented in the scalar case, and then extended into the vectorial framework. The main features and novelty of the paper are the (p, q) growth of the fractional operator, the double lack of compactness as well as the fact that the systems can be degenerate. As far as we know the results are new even in the scalar case and when the Kirchhoff model considered is non–degenerate.

scholarly journals Existence of Solutions for Choquard Type Elliptic Problems with Doubly Critical Nonlinearities

2019 ◽  
Vol 21 (1) ◽  
pp. 77-93
Author(s):  
Yansheng Shen
Keyword(s):  
Variational Methods ◽  
Minimization Problem ◽  
Existence Of Solutions ◽  
Elliptic Problems ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Critical Nonlinearities

Abstract In this article, we first study the existence of nontrivial solutions to the nonlocal elliptic problems in ℝ N {\mathbb{R}^{N}} involving fractional Laplacians and the Hardy–Sobolev–Maz’ya potential. Using variational methods, we investigate the attainability of the corresponding minimization problem, and then obtain the existence of solutions. We also consider another Choquard type equation involving the p-Laplacian and critical nonlinearities in ℝ N {\mathbb{R}^{N}} .

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.

scholarly journals Critical nonlocal Schrödinger-Poisson system on the Heisenberg group

2021 ◽  
Vol 11 (1) ◽  
pp. 482-502
Author(s):  
Zeyi Liu ◽  
Lulu Tao ◽  
Deli Zhang ◽  
Sihua Liang ◽  
Yueqiang Song
Keyword(s):  
Heisenberg Group ◽  
Mountain Pass Theorem ◽  
Poisson System ◽  
Heisenberg Groups ◽  
Euclidean Case ◽  
Smooth Bounded Domain ◽  
Critical Point Theorem ◽  
Critical Nonlinearities ◽  
Existence And Multiplicity ◽  
Non Local

Abstract In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: − a − b ∫ Ω | ∇ H u | 2 d ξ Δ H u + μ ϕ u = λ | u | q − 2 u + | u | 2 u , in Ω , − Δ H ϕ = u 2 , in Ω , u = ϕ = 0 , on ∂ Ω , $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega,\\ u=\phi=0,\quad &\mbox{on}\, \partial\Omega, \end{cases} \end{equation*}$$ where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 $ \mathbb{H}^1 $ , and Ω ⊂ H 1 $ \Omega\subset \mathbb{H}^1 $ is a smooth bounded domain, a, b > 0, 1 < q < 2 or 2 < q < 4, λ > 0 and μ ∈ R $ \mu\in \mathbb{R} $ are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient (a − b∫Ω∣∇ H u∣2 dx) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.

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 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 Kirchhoff–Hardy Fractional Problems with Lack of Compactness

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Alessio Fiscella ◽  
Patrizia Pucci
Keyword(s):  
Asymptotic Behavior ◽  
Differential Operator ◽  
Nontrivial Solutions ◽  
Hardy Potential ◽  
Lack Of Compactness ◽  
Critical Nonlinearities

AbstractThis paper deals with the existence and the asymptotic behavior of nontrivial solutions for some classes of stationary Kirchhoff problems driven by a fractional integro-differential operator and involving a Hardy potential and different critical nonlinearities. In particular, we cover the delicate

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

Fractional problems with critical nonlinearities by a sublinear perturbation

2020 ◽  
Vol 23 (2) ◽  
pp. 484-503 ◽  
Author(s):  
Lin Li ◽  
Stepan Tersian
Keyword(s):  
Critical Exponent ◽  
Variational Approach ◽  
Local Minimum ◽  
Numerical Estimate ◽  
Nontrivial Solutions ◽  
The Real ◽  
Critical Nonlinearities ◽  
Minimum Theorem ◽  
Sublinear Perturbation

AbstractIn this paper, the existence of two nontrivial solutions for a fractional problem with critical exponent, depending on real parameters, is established. The variational approach is used based on a local minimum theorem due to G. Bonanno. In addition, a numerical estimate on the real parameters is provided, for which the two solutions are obtained.

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