A Generalized BEC System with Critical Exponents in Dimensions $$N\ge 5$$

2014 ◽  
pp. 97-156
Author(s):  
Zhijie Chen
Keyword(s):  
Critical Exponents

Critical exponents for Ising-like systems on Sierpinski carpets

1987 ◽  
Vol 48 (4) ◽  
pp. 553-558 ◽  
Author(s):  
B. Bonnier ◽  
Y. Leroyer ◽  
C. Meyers
Keyword(s):  
Critical Exponents ◽  
Sierpinski Carpets

scholarly journals Critical exponents from five-loop scalar theory renormalization near six-dimensions

2021 ◽  
Vol 817 ◽  
pp. 136331
Author(s):  
Mikhail Kompaniets ◽  
Andrey Pikelner
Keyword(s):  
Critical Exponents ◽  
Scalar Theory ◽  
Six Dimensions

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.

scholarly journals Critical exponents in coupled phase-oscillator models on small-world networks

2020 ◽  
Vol 102 (6) ◽  
Author(s):  
Ryosuke Yoneda ◽  
Kenji Harada ◽  
Yoshiyuki Y. Yamaguchi
Keyword(s):  
Critical Exponents ◽  
Small World ◽  
Phase Oscillator ◽  
Small World Networks ◽  
Coupled Phase Oscillator ◽  
Oscillator Models

scholarly journals High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents

2021 ◽  
Vol 287 ◽  
pp. 329-375
Author(s):  
Fashun Gao ◽  
Haidong Liu ◽  
Vitaly Moroz ◽  
Minbo Yang
Keyword(s):  
Positive Solutions ◽  
Critical Exponents ◽  
High Energy

scholarly journals Quantum critical exponents for a disordered three-dimensional Weyl node

2015 ◽  
Vol 92 (11) ◽  
Author(s):  
Björn Sbierski ◽  
Emil J. Bergholtz ◽  
Piet W. Brouwer
Keyword(s):  
Critical Exponents ◽  
Three Dimensional ◽  
Quantum Critical
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