scholarly journals Existence and Symmetry of Solutions for a Class of Fractional Schrödinger–Poisson Systems

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1149
Author(s):  
Yongzhen Yun ◽  
Tianqing An ◽  
Guoju Ye
Keyword(s):  
Variational Methods ◽  
Critical Growth ◽  
Concentration Compactness ◽  
Radially Symmetric Solutions

In this paper, we investigate a class of Schrödinger–Poisson systems with critical growth. By the principle of concentration compactness and variational methods, we prove that the system has radially symmetric solutions, which improve the related results on this topic.

scholarly journals Radially symmetric solutions to the Hénon–Lane–Emden system on the critical hyperbola

2014 ◽  
Vol 16 (03) ◽  
pp. 1350030 ◽  
Author(s):  
Roberta Musina ◽  
K. Sreenadh
Keyword(s):  
Variational Methods ◽  
Existence Results ◽  
Qualitative Properties ◽  
Radially Symmetric Solutions ◽  
Qualitative Properties Of Solutions

We use variational methods to study the existence of non-trivial and radially symmetric solutions to the Hénon–Lane–Emden system with weights, when the exponents involved lie on the "critical hyperbola". We also discuss qualitative properties of solutions and non-existence results.

EXISTENCE AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS FOR DEGENERATE HEMIVARIATIONAL INEQUALITIES INVOLVING LERAY–LIONS TYPE OPERATOR WITH CRITICAL GROWTH

2013 ◽  
Vol 11 (01) ◽  
pp. 1350007
Author(s):  
KAIMIN TENG
Keyword(s):  
Critical Growth ◽  
Type Operator ◽  
Hemivariational Inequalities ◽  
Concentration Compactness ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Concentration Compactness Principle ◽  
Nonsmooth Critical Point ◽  
Critical Point Theorems ◽  
Compactness Principle

In this paper, we investigate a hemivariational inequality involving Leray–Lions type operator with critical growth. Some existence and multiple results are obtained through using the concentration compactness principle of P. L. Lions and some nonsmooth critical point theorems.

scholarly journals The Brezis–Nirenberg problem for nonlocal systems

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Luiz F. O. Faria ◽  
Olimpio H. Miyagaki ◽  
Fabio R. Pereira ◽  
Marco Squassina ◽  
Chengxiang Zhang
Keyword(s):  
Variational Methods ◽  
Weak Solutions ◽  
Fractional Laplacian ◽  
Critical Growth ◽  
Laplacian Operator ◽  
Nonlocal Equations ◽  
Suitable Sense ◽  
Regularity Of Weak Solutions ◽  
Fractional Laplacian Operator ◽  
Nirenberg Problem

AbstractBy means of variational methods we investigate existence, nonexistence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

scholarly journals Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth

2018 ◽  
Vol 8 (1) ◽  
pp. 1184-1212 ◽  
Author(s):  
Daniele Cassani ◽  
Jianjun Zhang
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Sobolev Inequality ◽  
Ground States ◽  
Critical Growth ◽  
Qualitative Behavior ◽  
Schrödinger Equations ◽  
Qualitative Properties ◽  
Concentration Phenomena ◽  
Qualitative Properties Of Solutions

Abstract We are concerned with the existence of ground states and qualitative properties of solutions for a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits critical growth in the sense of the Hardy–Littlewood–Sobolev inequality, in the range of the so-called upper-critical exponent. Qualitative behavior and concentration phenomena of solutions are also studied. Our approach turns out to be robust, as we do not require the nonlinearity to enjoy monotonicity nor Ambrosetti–Rabinowitz-type conditions, still using variational methods.

Multiple solutions for a class of p ( x )-Laplacian equations in involving the critical exponent

2010 ◽  
Vol 466 (2118) ◽  
pp. 1667-1686 ◽  
Author(s):  
Yongqiang Fu ◽  
Xia Zhang
Keyword(s):  
Critical Exponent ◽  
Multiple Solutions ◽  
Existence Of Solutions ◽  
Concentration Compactness ◽  
Concentration Compactness Principle ◽  
Radially Symmetric Solutions ◽  
Compactness Principle ◽  
Laplacian Equations

In this paper, we first establish a principle of concentration compactness in . Then, based on this concentration compactness principle, we study the existence of solutions for a class of p ( x )-Laplacian equations in involving the critical exponent. Under suitable assumptions, we obtain a sequence of radially symmetric solutions associated with a sequence of positive energies going towards infinity.

scholarly journals A system of Schrödinger equations with general quadratic-type nonlinearities

2020 ◽  
pp. 2050023
Author(s):  
Norman Noguera ◽  
Ademir Pastor
Keyword(s):  
Global Existence ◽  
Variational Methods ◽  
Nonlinear Stability ◽  
Ground State ◽  
Finite Time ◽  
Blow Up ◽  
Quadratic Growth ◽  
Schrodinger Equations ◽  
Concentration Compactness ◽  
Schrödinger Equations

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.

scholarly journals EXISTENCE OF SOLUTIONS FOR CRITICAL SYSTEMS WITH VARIABLE EXPONENTS

2018 ◽  
Vol 23 (4) ◽  
pp. 596-610 ◽  
Author(s):  
Hadjira Lalilia ◽  
Saadia Tas ◽  
Ali Djellit
Keyword(s):  
Existence Result ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Growth Conditions ◽  
Critical Growth ◽  
Concentration Compactness ◽  
Variable Exponents ◽  
Critical Systems ◽  
Concentration Compactness Principle ◽  
Compactness Principle

In this work, we deal with elliptic systems under critical growth conditions on the nonlinearities. Using a variant of concentration-compactness principle, we prove an existence result.

scholarly journals Ground state solutions for Kirchhoff-type equations with a general nonlinearity in the critical growth

2018 ◽  
Vol 7 (4) ◽  
pp. 535-546 ◽  
Author(s):  
Li-Ping Xu ◽  
Haibo Chen
Keyword(s):  
Variational Methods ◽  
Ground State ◽  
Critical Growth ◽  
Ground State Solutions ◽  
Kirchhoff Type ◽  
General Nonlinearity

AbstractIn this paper, we concern ourselves with the following Kirchhoff-type equations:\left\{\begin{aligned} \displaystyle-\biggl{(}a+b\int_{\mathbb{R}^{3}}\lvert% \nabla u\rvert^{2}\,dx\biggr{)}\triangle u+Vu&\displaystyle=f(u)\quad\text{in % }\mathbb{R}^{3},\\ \displaystyle u&\displaystyle\in H^{1}(\mathbb{R}^{3}),\end{aligned}\right.where a, b and V are positive constants and f has critical growth. We use variational methods to prove the existence of ground state solutions. In particular, we do not use the classical Ambrosetti–Rabinowitz condition. Some recent results are extended.

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.

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