Abstract:This paper is concerned with the nonlocal elliptic system driven by the variable-order fractional magnetic Laplace operator involving concave-convex nonlinearities\begin{equation*}\left\{\begin{array}{rl}(-\Delta)_{A}^{s(\cdot)} u&=\lambda~ a(x)| u|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x)+\beta(x)}c(x)|u|^{\alpha(x)-2}u| v| ^{\beta(x)}, \hspace{2mm}{\rm in}\ \Omega, \\(-\Delta)_{A}^{s(\cdot)} v&=\mu~ b(x)| v|^{q(x)-2}v+\frac{\beta(x)}{\alpha(x)+\beta(x)}c(x)| u|^{\alpha(x)}| v| ^{\beta(x)-2}v, \hspace{2.5mm}{\rm in}\ \Omega, \\u=v&=0 , \hspace{1cm} {\rm in}\ \mathbb{R}^N\backslash\Omega,\end{array}\right.\end{equation*}where $\Omega\subset\mathbb R^N, ~N\geq2$ is a smooth bounded domain, $\lambda, \mu>0$ are the parameters,$s\in C(\mathbb R^N\times \mathbb R^N, (0, 1))$ and $q, \alpha, \beta\in C(\overline{\Omega}, (1, \infty))$ are the variable exponents and$a, b, c\in C(\overline{\Omega}, [0, \infty))$ are the non-negative weight functions. $(-\Delta)_{A}^{s(\cdot)}$ is the variable-order fractional magnetic Laplace operator, the magnetic field is $A\in C^{0, \alpha}(\mathbb R^N, \mathbb R^N)$ with $\alpha\in(0, 1]$ and $u:\mathbb R^N\to\mathbb C$. Use Nehari manifold to prove that there exists $\Lambda>0$ such that $\forall\lambda+\mu<\Lambda$, this system obtains at least two non-negative solutions of theabove problem under some assumptions on $q, \alpha, \beta$.

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	1. Nehari manifold for fractional s($\cdot$)-Laplacian system involving concave-convex nonlinearities with magnetic field
	Feng Dong-Xue,Chen Wen-Jing
	Mathematics 07 June 2023
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	Abstract:This paper is concerned with the nonlocal elliptic system driven by the variable-order fractional magnetic Laplace operator involving concave-convex nonlinearities\begin{equation}\left\{\begin{array}{rl}(-\Delta)_{A}^{s(\cdot)} u&=\lambda~ a(x)\| u\|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x)+\beta(x)}c(x)\|u\|^{\alpha(x)-2}u\| v\| ^{\beta(x)}, \hspace{2mm}{\rm in}\ \Omega, \\(-\Delta)_{A}^{s(\cdot)} v&=\mu~ b(x)\| v\|^{q(x)-2}v+\frac{\beta(x)}{\alpha(x)+\beta(x)}c(x)\| u\|^{\alpha(x)}\| v\| ^{\beta(x)-2}v, \hspace{2.5mm}{\rm in}\ \Omega, \\u=v&=0 , \hspace{1cm} {\rm in}\ \mathbb{R}^N\backslash\Omega,\end{array}\right.\end{equation}where $\Omega\subset\mathbb R^N, ~N\geq2$ is a smooth bounded domain, $\lambda, \mu>0$ are the parameters,$s\in C(\mathbb R^N\times \mathbb R^N, (0, 1))$ and $q, \alpha, \beta\in C(\overline{\Omega}, (1, \infty))$ are the variable exponents and$a, b, c\in C(\overline{\Omega}, [0, \infty))$ are the non-negative weight functions. $(-\Delta)_{A}^{s(\cdot)}$ is the variable-order fractional magnetic Laplace operator, the magnetic field is $A\in C^{0, \alpha}(\mathbb R^N, \mathbb R^N)$ with $\alpha\in(0, 1]$ and $u:\mathbb R^N\to\mathbb C$. Use Nehari manifold to prove that there exists $\Lambda>0$ such that $\forall\lambda+\mu<\Lambda$, this system obtains at least two non-negative solutions of theabove problem under some assumptions on $q, \alpha, \beta$.
	TO cite this article:Feng Dong-Xue,Chen Wen-Jing. Nehari manifold for fractional s($\cdot$)-Laplacian system involving concave-convex nonlinearities with magnetic field[OL].[ 7 June 2023] http://en.paper.edu.cn/en_releasepaper/content/4760785


	2. Existence and stability of traveling waves for nonlinear Schr\"odinger equations with van der Waals type potentials
	Lu Hui, Wu Dan
	Mathematics 11 May 2023
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	Abstract:In this paper, we study the existence and some stability results of traveling wave solutions of the semi-pseudo-relativistic Schr\"odinger equation with van der Waals potential. Based on the variational method, we study the corresponding constraint minimization problem by using the principle of concentrated compactness, and deduce the existence of the global minimizer of the minimization problem, thus obtaining the existence of the boosted ground state of Euler-Lagrange equaton. Furthermore, it is proved that the traveling wave solutions are orbitally stable.
	TO cite this article:Lu Hui, Wu Dan. Existence and stability of traveling waves for nonlinear Schr\"odinger equations with van der Waals type potentials[OL].[11 May 2023] http://en.paper.edu.cn/en_releasepaper/content/4760739

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	3. Ramsey numbers of multiple copies of graphs in a component
	HUANG CaiXia,PENG YueJian
	Mathematics 09 May 2023
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	Abstract:For a graph $G$, $R(c(nG))$ denotes the least positive integer $N$ such that every 2-colouring of the edges of $K_N$ contains a copy of $nG$ in a monochromatic component, where $nG$ denotes the graph consisting of $n$ vertex disjoint copies of $G$.Gy\'{a}f\'{a}s and S\'{a}rk\"{o}zy showed that $R(c(nK_3))=7n-2$ for $n \geq 2$ in 2016.After that, Roberts showed that $R(c(nK_r))=(r^2-r+1)n-r+1$ for $r \geq 4$ and $n \geq R(K_r)$ in 2017.This paper determines the values of $R(c(n(K_{1,3}+e)))$ and $R(c(n(K_4-e)))$.
	TO cite this article:HUANG CaiXia,PENG YueJian. Ramsey numbers of multiple copies of graphs in a component[OL].[ 9 May 2023] http://en.paper.edu.cn/en_releasepaper/content/4760562


	4. Boundedness and compactness of multilinear singular integrals on Morrey spaces
	MEI Ting,LI Ao-Bo
	Mathematics 20 April 2023
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	Abstract:In this paper, we consider the boundedness and compactness of the multilinear singular integral operator on Morrey spaces, which is defined by\begin{align}T_Af(x)={\rm{p.v.}}\int_{\mathbb{R}^n} \frac{\Omega(x-y)}{\|x-y\|^{n+1}} R(A;x,y)f(y)dy,\end{align}where $R(A;x,y)=A(x)-A(y)-\nabla A(y)\cdot(x-y)$ with $D^\beta A\in BMO(\mathbb{R}^n)$ for all $\|\beta\|=1$.We prove that $T_A$ is bounded and compact on Morrey spaces $L^{p,\lambda}(\mathbb{R}^n)$ for all $1<p<\infty$ with $\Omega$ and $A$ satisfying some conditions. Moreover, the boundedness and compactness of the maximal multilinear singular integral operator $T_{A,*}$ on Morrey spaces are also given in this paper.
	TO cite this article:MEI Ting,LI Ao-Bo. Boundedness and compactness of multilinear singular integrals on Morrey spaces[OL].[20 April 2023] http://en.paper.edu.cn/en_releasepaper/content/4760244


	5. Analysis and comparison of stabilized multiscale finite volume iterative schemes for the steady incompressible Navier-Stokes equations
	JIANG Yu-Lei,CHEN Chuan-Jun
	Mathematics 11 April 2023
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	Abstract:In this paper, three finite volume iterative schemes for the steady incompressible Navier-Stokes equations are provided based on the multiscale enrichment method.Under different restriction on the viscosity parameter, the stability and convergence results of the considered numerical schemes are established. Theoretical findings show that the Stokes and Newton iterations are stable under some strong uniqueness conditions, while the Oseen iteration is unconditionally stable and convergent under the uniqueness condition. Furthermore, the Newton iteration is exponential convergence with respect to the iterative step. Numerical examples are presented to verify the established theoretical findings and show the performances of three iterative finite volume methods.
	TO cite this article:JIANG Yu-Lei,CHEN Chuan-Jun. Analysis and comparison of stabilized multiscale finite volume iterative schemes for the steady incompressible Navier-Stokes equations[OL].[11 April 2023] http://en.paper.edu.cn/en_releasepaper/content/4759726


	6. The multiscale enrichment finite volumemethod for the stationary incompressible magnetohydrodynamics flow
	SHEN Xiao-Rong,ZHANG Tong,CHEN Chuan-Jun
	Mathematics 10 April 2023
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	Abstract:In the paper, the finite volume method for the steady incompressible magnetohydrodynamics(MHD) problem based on the lowest order mixed finite element pair on triangular mesh is considered, the linear polynomial is used to approximate the velocity and magnetic fields and the piecewise constant is adopted to approximate the pressure. In order to overcome the restriction of discrete inf-sup (LBB) condition, the multiscale enrichment method is employed. Firstly, the existence, uniqueness and stability of numerical solutions are established through the fixed point theorem. Then, the optimal error estimates of numerical solutions in H1 and L2-norms are presented. Finally, some numerical results are provided to verify the established theoretical findings and show the performances of considered numerical scheme.
	TO cite this article:SHEN Xiao-Rong,ZHANG Tong,CHEN Chuan-Jun. The multiscale enrichment finite volumemethod for the stationary incompressible magnetohydrodynamics flow[OL].[10 April 2023] http://en.paper.edu.cn/en_releasepaper/content/4759648


	7. Existence of the multi-peak solutions for a nonlinear fractional Kirchhoff equation withmagnetic fields
	SUN Qian,CHEN Wenjing
	Mathematics 29 March 2023
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	Abstract:The aim of this paper focuses on the following class of fractional magnetic Kirchhofftype equation$$\left(a\varepsilon^{2s}+b\varepsilon^{4s-N}[u]_{A/\varepsilon}^2\right) (-\Delta)_{A/\varepsilon}^su+V(x)u=\|u\|^{p-1}u,\quad \mbox{in } \mathbb{R}^N,$$where $\varepsilon>0$ is a small parameter, $a,b>0$, $(-\Delta)_{A}^{s}$ is the fractional magneticLaplacian operator, $s\in (0,1)$, $p\in(1,\frac{N+2s}{N-2s})$ and $N\in(2s,4s)$, $A(x): \mathbb{R}^N\rightarrow \mathbb{R}^N$ is the bounded magnetic potential, $V(x): \mathbb{R}^N \rightarrow\mathbb{R}$ is a continuous potential function. Our approach is based on some decaying estimateand nondegenerate, we prove that the equation has multi-peak solutions concentrating at localminimum points of $V(x)$ by applying Lyapunov-Schmidt reduction method.
	TO cite this article:SUN Qian,CHEN Wenjing. Existence of the multi-peak solutions for a nonlinear fractional Kirchhoff equation withmagnetic fields[OL].[29 March 2023] http://en.paper.edu.cn/en_releasepaper/content/4759898


	8. Existence of ground state solutions for coupled Choquard system with lower critical exponents
	WANG Fen-Fen,DENG Sheng-Bing
	Mathematics 29 March 2023
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	Abstract:In this paper, we study the following coupled Choquard type system with Hardy--Littlewood--Sobolev lower critical exponents and a local nonlinear perturbation:\begin{equation}\left\{ \arraycolsep=1.5pt \begin{array}{ll}-\Delta u+V(x)u=\big(I_\alpha\|u\|^{\frac{\alpha}{N}+1}\big)\|u\|^{\frac{\alpha}{N}-1}u+ \lambda(I_\alpha\|v\|^{p})\|u\|^{p-2}u, &\ \text{ in } \mathbb{R}^N,\\-\Delta v+V(x)v=\big(I_\alpha\|v\|^{\frac{\alpha}{N}+1}\big)\|v\|^{\frac{\alpha}{N}-1}v+ \lambda(I_\alpha\|u\|^{p})\|v\|^{p-2}v, &\ \text{ in } \mathbb{R}^N,\\ \end{array} \right.\end{equation}where $N\geq 3$, $ \alpha \in (0,N)$, $I_{\alpha}:\mathbb{R}^N\backslash{\{0\}}\to\mathbb{R}$ is a Riesz potential,$V\in C(\mathbb{R}^N,[0,\infty))$ and satisfies some suitable conditions. In the case when $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N+1}$, $p=\frac{N+\alpha+2}{N+1}$, and $\frac{N+\alpha+2}{N+1}<p<\frac{N+\alpha}{N-2}$, respectively, we investigate the existence of positive ground states of this system if $\lambda>\lambda_{*}$ by variational approaches.
	TO cite this article:WANG Fen-Fen,DENG Sheng-Bing. Existence of ground state solutions for coupled Choquard system with lower critical exponents[OL].[29 March 2023] http://en.paper.edu.cn/en_releasepaper/content/4759877


	9. Lattice paths and the Prouhet-Thue-Morse sequence
	Bing Gao,Shishuo Fu
	Mathematics 16 March 2023
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	Abstract:Using Flajolet's combinatorial theory of continued fractions and axial symmetry of lattice paths, we show that the enumeration of certain lattice paths modulo two yields the ubiquitous Prouhet-Thue-Morse sequence. This answers an open problem of Berstel et al.
	TO cite this article:Bing Gao,Shishuo Fu. Lattice paths and the Prouhet-Thue-Morse sequence[OL].[16 March 2023] http://en.paper.edu.cn/en_releasepaper/content/4759531


	10. Measure-theoretic Entropy for Weak-solvable Cancellative Left-amenable Semigroup
	HUANG ShiYao,HUANG XiaoJun
	Mathematics 16 March 2023
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	Abstract:The study of dynamical systems under semigroup actions is an important branch of topological dynamical systems. At the same time, the study of dynamical system entropy is also of great significance, among which metric entropy can measure the complexity of the motion of a dynamical system on a probability space and constitutes an invariant of isomorphic systems. The main research content of this article is to generalize the Fekete lemma to weakly solvable cancellative conformal semigroups, and prove that the limit related to the F$\phi$lner sequence in this semigroup satisfies the "infimum rule". Finally, the metric entropy of the dynamical system under this semigroup action is given.
	TO cite this article:HUANG ShiYao,HUANG XiaoJun. Measure-theoretic Entropy for Weak-solvable Cancellative Left-amenable Semigroup[OL].[16 March 2023] http://en.paper.edu.cn/en_releasepaper/content/4759489

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