On the Critical Exponent for k-Primitive Sets

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.

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

Advances in Nonlinear Analysis ◽

10.1515/anona-2020-0151 ◽

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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Crossover in the dynamical critical exponent of a quenched two-dimensional Bose gas

Physical Review Research ◽

10.1103/physrevresearch.3.013212 ◽

2021 ◽

Vol 3 (1) ◽

Author(s):

A. J. Groszek ◽

P. Comaron ◽

N. P. Proukakis ◽

T. P. Billam

Keyword(s):

Critical Exponent ◽

Bose Gas ◽

Two Dimensional

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Critical exponent for nonlinear damped wave equations with non-negative potential in 3D

Journal of Differential Equations ◽

10.1016/j.jde.2019.04.004 ◽

2019 ◽

Vol 267 (5) ◽

pp. 3271-3288

Author(s):

Vladimir Georgiev ◽

Hideo Kubo ◽

Kyouhei Wakasa

Keyword(s):

Critical Exponent ◽

Wave Equations ◽

Negative Potential ◽

Nonlinear Damped Wave Equations

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Existence and Convergence of Solutions to Fractional Pure Critical Exponent Problems

Advanced Nonlinear Studies ◽

10.1515/ans-2021-2041 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Víctor Hernández-Santamaría ◽

Alberto Saldaña

Keyword(s):

Critical Exponent ◽

Convergence Properties ◽

Scaling Invariance ◽

Convergence Of Solutions ◽

Nonexistence Result ◽

Homogeneous Sobolev Space ◽

Similar Characterization ◽

Nonnegative Solutions ◽

Least Energy ◽

Local Transition

Abstract We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem ( - Δ ) s ⁢ u s = | u s | 2 s ⋆ - 2 ⁢ u s , u s ∈ D 0 s ⁢ ( Ω ) , 2 s ⋆ := 2 ⁢ N N - 2 ⁢ s , (-\Delta)^{s}u_{s}=\lvert u_{s}\rvert^{2_{s}^{\star}-2}u_{s},\quad u_{s}\in D^% {s}_{0}(\Omega),\,2^{\star}_{s}:=\frac{2N}{N-2s}, where s is any positive number, Ω is either ℝ N {\mathbb{R}^{N}} or a smooth symmetric bounded domain, and D 0 s ⁢ ( Ω ) {D^{s}_{0}(\Omega)} is the homogeneous Sobolev space. Depending on the kind of symmetry considered, solutions can be sign-changing. We show that, up to a subsequence, a l.e.s.s. u s {u_{s}} converges to a l.e.s.s. u t {u_{t}} as s goes to any t > 0 {t>0} . In bounded domains, this convergence can be characterized in terms of an homogeneous fractional norm of order t - ε {t-\varepsilon} . A similar characterization is no longer possible in unbounded domains due to scaling invariance and an incompatibility with the functional spaces; to circumvent these difficulties, we use a suitable rescaling and characterize the convergence via cut-off functions. If t is an integer, then these results describe in a precise way the nonlocal-to-local transition. Finally, we also include a nonexistence result of nontrivial nonnegative solutions in a ball for any s > 1 {s>1} .

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Renormalization group theory of molecular dynamics

Scientific Reports ◽

10.1038/s41598-021-85286-3 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Daiji Ichishima ◽

Yuya Matsumura

Keyword(s):

Molecular Dynamics ◽

Renormalization Group ◽

Critical Exponent ◽

Computational Efficiency ◽

Large Scale ◽

Dissipative Particle Dynamics ◽

Computational Cost ◽

Coarse Graining ◽

Spatial Dimension ◽

Deborah Number

AbstractLarge scale computation by molecular dynamics (MD) method is often challenging or even impractical due to its computational cost, in spite of its wide applications in a variety of fields. Although the recent advancement in parallel computing and introduction of coarse-graining methods have enabled large scale calculations, macroscopic analyses are still not realizable. Here, we present renormalized molecular dynamics (RMD), a renormalization group of MD in thermal equilibrium derived by using the Migdal–Kadanoff approximation. The RMD method improves the computational efficiency drastically while retaining the advantage of MD. The computational efficiency is improved by a factor of $$2^{n(D+1)}$$ 2 n ( D + 1 ) over conventional MD where D is the spatial dimension and n is the number of applied renormalization transforms. We verify RMD by conducting two simulations; melting of an aluminum slab and collision of aluminum spheres. Both problems show that the expectation values of physical quantities are in good agreement after the renormalization, whereas the consumption time is reduced as expected. To observe behavior of RMD near the critical point, the critical exponent of the Lennard-Jones potential is extracted by calculating specific heat on the mesoscale. The critical exponent is obtained as $$\nu =0.63\pm 0.01$$ ν = 0.63 ± 0.01 . In addition, the renormalization group of dissipative particle dynamics (DPD) is derived. Renormalized DPD is equivalent to RMD in isothermal systems under the condition such that Deborah number $$De\ll 1$$ D e ≪ 1 .

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Existence and multiplicity of solutions for Kirchhof-type problems with Sobolev–Hardy critical exponent

Boundary Value Problems ◽

10.1186/s13661-021-01521-w ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Hongsen Fan ◽

Zhiying Deng

Keyword(s):

Variational Method ◽

Boundary Value Problem ◽

Critical Exponent ◽

Positive Solution ◽

Elliptic Boundary ◽

Boundary Value ◽

Elliptic Boundary Value Problem ◽

Nontrivial Solutions ◽

Elliptic Boundary Value ◽

Existence And Multiplicity

AbstractIn this paper, we discuss a class of Kirchhof-type elliptic boundary value problem with Sobolev–Hardy critical exponent and apply the variational method to obtain one positive solution and two nontrivial solutions to the problem under certain conditions.

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