scholarly journals A Remark on the Concentration Compactness Principle in Critical Dimension

2021 ◽  
Author(s):  
Fengbo Hang
Keyword(s):  
Critical Dimension ◽  
Concentration Compactness ◽  
Concentration Compactness Principle ◽  
Compactness Principle

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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