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.

Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation

2020 ◽  
Vol 10 (1) ◽  
pp. 131-151
Author(s):  
Yueli Liu ◽  
Xu Li ◽  
Chao Ji
Keyword(s):  
Variational Methods ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Concentrating Solutions ◽  
Concentration Properties ◽  
Penalization Technique ◽  
Poisson Type

Abstract In this paper, we study the following nonlinear magnetic Schrödinger-Poisson type equation $$\begin{array}{} \displaystyle \left\{\!\begin{aligned}&\Big(\frac{\varepsilon}{i}\nabla-A(x)\Big)^{2}u+V(x)u+\epsilon^{-2}(\vert x\vert^{-1}\ast \vert u\vert^{2})u = f(|u|^{2})u\quad\hbox{in }\mathbb{R}^3,\\&u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}),\end{aligned}\right. \end{array}$$ where ϵ > 0, V : ℝ3 → ℝ and A : ℝ3 → ℝ3 are continuous potentials. Under a local assumption on the potential V, by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration properties of nontrivial solutions for ε > 0 small. In this problem, the function f is only continuous, which allow to consider larger classes of nonlinearities in the reaction.

scholarly journals Multiplicity and concentration results for a fractional Schrödinger-Poisson type equation with magnetic field

2019 ◽  
Vol 150 (2) ◽  
pp. 655-694 ◽  
Author(s):  
Vincenzo Ambrosio
Keyword(s):  
Magnetic Field ◽  
Variational Methods ◽  
Electric Potential ◽  
Nonlinear Term ◽  
Type Equation ◽  
Magnetic Potential ◽  
Nontrivial Solutions ◽  
Image Position ◽  
Poisson Type

AbstractThis paper is devoted to the study of fractional Schrödinger-Poisson type equations with magnetic field of the type $$\varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u + V(x)u + {\rm e}^{-2t}(\vert x \vert^{2t-3} \ast \vert u\vert ^{2})u = f(\vert u \vert^{2})u \quad \hbox{in} \ \open{R}^{3},$$ where ε > 0 is a parameter, s, t ∈ (0, 1) are such that 2s+2t>3, A:ℝ3 → ℝ3 is a smooth magnetic potential, (−Δ)As is the fractional magnetic Laplacian, V:ℝ3 → ℝ is a continuous electric potential and f:ℝ → ℝ is a C1 subcritical nonlinear term. Using variational methods, we obtain the existence, multiplicity and concentration of nontrivial solutions for e > 0 small enough.

scholarly journals Multiple positive bound states for critical Schrödinger-Poisson systems

2019 ◽  
Vol 25 ◽  
pp. 73 ◽  
Author(s):  
Giovanna Cerami ◽  
Riccardo Molle
Keyword(s):  
Variational Methods ◽  
Ground State ◽  
Bound States ◽  
Nonlinear Term ◽  
Critical Growth ◽  
Poisson System ◽  
Lack Of Compactness ◽  
Ground State Solutions ◽  
Existence And Multiplicity

Using variational methods we prove some results about existence and multiplicity of positive bound states of to the following Schrödinger-Poisson system: [see formula in PDF] We remark that (SP) exhibits a “double” lack of compactness because of the unboundedness of ℝ3 and the critical growth of the nonlinear term and that in our assumptions ground state solutions of (SP) do not exist.

Existence and multiplicity of solutions for Kirchhoff–Schrödinger–Poisson system with critical growth

2021 ◽  
Author(s):  
Guofeng Che ◽  
Haibo Chen
Keyword(s):  
Variational Methods ◽  
Critical Growth ◽  
Poisson System ◽  
Nontrivial Solutions ◽  
Multiplicity Of Solutions ◽  
Existence And Multiplicity

This paper is concerned with the following Kirchhoff–Schrödinger–Poisson system: [Formula: see text] where constants [Formula: see text], [Formula: see text] and [Formula: see text] are the parameters. Under some appropriate assumptions on [Formula: see text], [Formula: see text] and [Formula: see text], we prove the existence and multiplicity of nontrivial solutions for the above system via variational methods. Some recent results from the literature are greatly improved and extended.

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}} .

Critical groups and multiple solutions for Kirchhoff type equations with critical exponents

2020 ◽  
pp. 2050031
Author(s):  
Mingzheng Sun ◽  
Jiabao Su ◽  
Binlin Zhang
Keyword(s):  
Critical Exponents ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Nonlinear Term ◽  
Type Equation ◽  
Critical Groups ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
The First Eigenvalue

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities.

scholarly journals Nontrivial Solutions for Time Fractional Nonlinear Schrödinger-Kirchhoff Type Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
N. Nyamoradi ◽  
Y. Zhou ◽  
E. Tayyebi ◽  
B. Ahmad ◽  
A. Alsaedi
Keyword(s):  
Variational Methods ◽  
Fractional Derivatives ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Nontrivial Solutions ◽  
Nonlinear Schrödinger ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Left And Right

We study the existence of solutions for time fractional Schrödinger-Kirchhoff type equation involving left and right Liouville-Weyl fractional derivatives via variational methods.

Existence and Multiplicity of Solutions for the Semi-Linear Dirichlet Problem with a Logarithmic Nonlinear Term

2014 ◽  
Vol 526 ◽  
pp. 177-181
Author(s):  
Yuan Li ◽  
Ai Hui Sheng
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Nonlinear Term ◽  
Mountain Pass Theorem ◽  
Nehari Manifold ◽  
Nontrivial Solutions ◽  
Perturbation Theorem ◽  
Existence And Multiplicity ◽  
Manifold Theory ◽  
The Nehari Manifold

The Dirichlet problem with logarithmic nonlinear term doesn't satisfy (A.R) condition. By using the variant mountain pass theorem and perturbation theorem of variational methods, the existence of nontrivial solutions are established for . We also introduce some deformation of equation with a logarithmic nonlinear term, the sign-changing solution, the Nehari manifold theory, bifurcation theory, improve the theory of variational methods.

Multiplicity and Concentration Results for a Magnetic Schrödinger Equation With Exponential Critical Growth in ℝ2

2020 ◽  
Author(s):  
Pietro d’Avenia ◽  
Chao Ji
Keyword(s):  
Magnetic Field ◽  
Schrödinger Equation ◽  
Variational Methods ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Critical Growth ◽  
Concentration Of Solutions ◽  
Exponential Critical Growth ◽  
Penalization Technique

Abstract In this paper we study the following nonlinear Schrödinger equation with magnetic field $$\begin{align*} \left(\frac{\varepsilon}{i}\nabla-A(x)\right)^{2}u+V(x)u=f(| u|^{2})u,\quad x\in\mathbb{R}^{2}, \end{align*}$$where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous potentials, and $f:\mathbb{R}\rightarrow \mathbb{R}$ has exponential critical growth. Under a local assumption on the potential $V$, by variational methods, penalization technique, and Ljusternik–Schnirelmann theory, we prove multiplicity and concentration of solutions for $\varepsilon $ small.

scholarly journals A strongly indefinite Choquard equation with critical exponent due to the Hardy–Littlewood–Sobolev inequality

2018 ◽  
Vol 20 (04) ◽  
pp. 1750037 ◽  
Author(s):  
Fashun Gao ◽  
Minbo Yang
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Nontrivial Solution ◽  
Sobolev Inequality ◽  
Schrodinger Equation ◽  
Critical Growth ◽  
Nontrivial Solutions ◽  
Choquard Equation ◽  
Semilinear Equation ◽  
Nonlinear Choquard Equation

In this paper, we are concerned with the following nonlinear Choquard equation [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text] lies in a gap of the spectrum of [Formula: see text] and [Formula: see text] is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557–6579].

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