scholarly journals EXISTENCE OF STEADY STATES FOR THE MAXWELL–SCHRÖDINGER–POISSON SYSTEM: EXPLORING THE APPLICABILITY OF THE CONCENTRATION–COMPACTNESS PRINCIPLE

2013 ◽  
Vol 23 (10) ◽  
pp. 1915-1938 ◽  
Author(s):  
I. CATTO ◽  
J. DOLBEAULT ◽  
O. SÁNCHEZ ◽  
J. SOLER
Keyword(s):  
Steady States ◽  
Concentration Compactness ◽  
Poisson System ◽  
Open Problems ◽  
Compactness Method ◽  
System A ◽  
Concentration Compactness Principle ◽  
Compactness Principle ◽  
Schrödinger System

This paper reviews recent results and open problems concerning the existence of steady states to the Maxwell–Schrödinger system. A combination of tools, proofs and results are presented in the framework of the concentration–compactness method.

scholarly journals Multiple solutions for a quasilinear Schrödinger system of Kirchhoff type with critical exponents

2018 ◽  
Vol 37 (4) ◽  
pp. 187-203
Author(s):  
Mohammed Massar ◽  
Ahmed Hamydy ◽  
Najib Tsouli
Keyword(s):  
Critical Exponents ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Type Systems ◽  
Concentration Compactness ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Genus Theory ◽  
Compactness Principle ◽  
Schrödinger System

This paper is devoted to the existence of solutions for a class of Kirchhoff type systems involving critical exponents. The proof of the main results is based on  concentration compactness principle related to critical elliptic systems due to Kang combined with genus theory.

scholarly journals Existence of positive ground state solutions of Schrödinger–Poisson system involving negative nonlocal term and critical exponent on bounded domain

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Wenxuan Zheng ◽  
Wenbin Gan ◽  
Shibo Liu
Keyword(s):  
Critical Exponent ◽  
Bounded Domain ◽  
Ground State ◽  
Mountain Pass Theorem ◽  
Nonlocal Term ◽  
Concentration Compactness ◽  
Poisson System ◽  
Ground State Solutions ◽  
Concentration Compactness Principle ◽  
Compactness Principle

AbstractIn this paper, we prove the existence of positive ground state solutions of the Schrödinger–Poisson system involving a negative nonlocal term and critical exponent on a bounded domain. The main tools are the mountain pass theorem and the concentration compactness principle.

scholarly journals Ground states for critical fractional Schr\”{o}dinger-Poisson systems with vanishing potentials

2021 ◽  
Author(s):  
Xilin Dou ◽  
xiaoming he
Keyword(s):  
Ground State ◽  
Mountain Pass Theorem ◽  
Ground State Solution ◽  
Nonlocal Term ◽  
Concentration Compactness ◽  
Poisson System ◽  
Concentration Compactness Principle ◽  
Whole Space ◽  
Compactness Principle ◽  
Vanishing Potentials

This paper deals with a class of fractional Schr\”{o}dinger-Poisson system \[\begin{cases}\displaystyle (-\Delta )^{s}u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=a (x)f(u), &x \in \R^{3}\\ (-\Delta )^{s}\phi=K(x)|u|^{2^*_s-1}, &x \in \R^{3}\end{cases} \]with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where $s \in (\frac{3}{4},1)$, $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical exponent. The problem is set on the whole space and compactness issues have to be tackled. By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on $V, K, a$ and $f$.

scholarly journals Multiple solutions for critical Choquard-Kirchhoff type equations

2020 ◽  
Vol 10 (1) ◽  
pp. 400-419 ◽  
Author(s):  
Sihua Liang ◽  
Patrizia Pucci ◽  
Binlin Zhang
Keyword(s):  
Critical Exponent ◽  
Critical Exponents ◽  
Sobolev Inequality ◽  
Multiple Solutions ◽  
Concentration Compactness ◽  
Positive Real ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Concentration Compactness Principle ◽  
Compactness Principle

Abstract In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, $$\begin{array}{} \displaystyle -\left(a + b\int\limits_{\mathbb{R}^N} |\nabla u|^2 dx\right){\it\Delta} u = \alpha k(x)|u|^{q-2}u + \beta\left(\,\,\displaystyle\int\limits_{\mathbb{R}^N}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u, \quad x \in \mathbb{R}^N, \end{array}$$ where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, $\begin{array}{} 2^*_{\mu} = (2N-\mu)/(N-2) \end{array}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting.

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.

Sign in / Sign up

Export Citation Format

Share Document