Concentration–Compactness principle for the sharp Adams inequalities in bounded domains and whole space Rn

2019 ◽  
Vol 267 (7) ◽  
pp. 4448-4492 ◽  
Author(s):  
Van Hoang Nguyen
Keyword(s):  
Bounded Domains ◽  
Concentration Compactness ◽  
Concentration Compactness Principle ◽  
Whole Space ◽  
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.

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