scholarly journals EXISTENCE AND CONCENTRATION RESULT FOR KIRCHHOFF EQUATIONS WITH CRITICAL EXPONENT AND HARTREE NONLINEARITY

2020 ◽  
Vol 10 (5) ◽  
pp. 2121-2144
Author(s):  
Guofeng Che ◽  
◽  
Haibo Chen ◽  
Keyword(s):  
Critical Exponent ◽  
Kirchhoff Equations ◽  
Concentration Result

scholarly journals Solutions to Kirchhoff equations with critical exponent

2016 ◽  
Vol 22 (1) ◽  
pp. 138-149 ◽  
Author(s):  
El Miloud Hssini ◽  
Mohammed Massar ◽  
Najib Tsouli
Keyword(s):  
Critical Exponent ◽  
Kirchhoff Equations

scholarly journals Correction to: Solving procedure for the Kelvin–Kirchhoff equations in case of non-stationary rotations of slim disc

2021 ◽  
Author(s):  
Sergey V. Ershkov ◽  
Dmytro Leshchenko ◽  
Ayrat R. Giniyatullin
Keyword(s):  
Kirchhoff Equations

Asymptotic behavior of global solutions to a class of mixed pseudo-parabolic Kirchhoff equations

2021 ◽  
Vol 118 ◽  
pp. 107119
Author(s):  
Yang Cao ◽  
Qiuting Zhao
Keyword(s):  
Asymptotic Behavior ◽  
Global Solutions ◽  
Kirchhoff Equations

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.

Infinitely many sign-changing solutions for nonlinear fractional Kirchhoff equations

2021 ◽  
pp. 1-22
Author(s):  
Guangze Gu ◽  
Xianyong Yang ◽  
Zhipeng Yang
Keyword(s):  
Kirchhoff Equations ◽  
Sign Changing Solutions

scholarly journals Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

2020 ◽  
Vol 10 (1) ◽  
pp. 732-774
Author(s):  
Zhipeng Yang ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Variational Methods ◽  
Critical Exponent ◽  
Laplace Operator ◽  
Sobolev Inequality ◽  
Fractional Laplacian ◽  
Singularly Perturbed ◽  
Choquard Equation ◽  
Fractional Choquard Equation ◽  
Fractional Laplace Operator

Abstract In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

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