scholarly journals A Pohožaev Identity and Critical Exponents of Some Complex Hessian Equations

2016 ◽  
Vol 29 (3) ◽  
pp. 175-194
Author(s):  
Li Chi
Keyword(s):  
Critical Exponents ◽  
Pohozaev Identity ◽  
Hessian Equations

scholarly journals On critical exponents of a k-Hessian equation in the whole space

2019 ◽  
Vol 149 (6) ◽  
pp. 1555-1575 ◽  
Author(s):  
Yun Wang ◽  
Yutian Lei
Keyword(s):  
Critical Exponents ◽  
Decay Rates ◽  
Classical Solutions ◽  
Sobolev Type ◽  
Extremal Functions ◽  
Hessian Equation ◽  
Hessian Equations ◽  
Radial Structure ◽  
Whole Space ◽  
Image Position

AbstractIn this paper, we study negative classical solutions and stable solutions of the following k-Hessian equation $$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$with radial structure, where n ⩾ 3, 1 < k < n/2 and p > 1. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research k-Hessian equations without radial structure.

scholarly journals Choquard equations with critical nonlinearities

2019 ◽  
Vol 22 (04) ◽  
pp. 1950023 ◽  
Author(s):  
Xinfu Li ◽  
Shiwang Ma
Keyword(s):  
Critical Exponents ◽  
Ground State ◽  
Sobolev Inequality ◽  
Riesz Potential ◽  
Ground State Solution ◽  
Type Problem ◽  
Pohozaev Identity ◽  
Choquard Equation ◽  
Critical Nonlinearities ◽  
Constraint Method

In this paper, we study the Brezis–Nirenberg type problem for Choquard equations in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text] are the critical exponents in the sense of Hardy–Littlewood–Sobolev inequality and [Formula: see text] is the Riesz potential. Based on the results of the subcritical problems, and by using the subcritical approximation and the Pohožaev constraint method, we obtain a positive and radially nonincreasing ground-state solution in [Formula: see text] for the problem. To the end, the regularity and the Pohožaev identity of solutions to a general Choquard equation are obtained.

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
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