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 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

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	3. Signed and sign-changing solutions for nonlinear Schr\"{o}dinger-Poisson system with two nonlocal terms in $\mathbb{R}^{3}$
	Yi Wen, Zeng-Qi Ou
	Mathematics 03 March 2023
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	Abstract:In this paper, we consider the following Schr\"{o}dinger-Poisson system \begin{equation} \left\{\begin{matrix} -\Delta u+V(x)u+\phi u=\left(\int_{\mathbb{R}^3}\frac{1}{p}\|u\|^pdx\right)^{\frac{2}{p}}\|u\|^{p-2}u+g(x)\|u\|^{q-2}u,& \mbox{in}\ \mathbb{R}^3,\\ -\Delta\phi=u^2,& \mbox{in}\ \mathbb{R}^3,\hfill\label{0.1} \end{matrix}\right. \end{equation} where $1<q<2<p<6$ and the functions $V(x), g(x)$ satisfy the certain conditions. Using variational methods and invariant sets of descending flow, we prove that system (\ref{0.1}) possesses three nontrivial solutions of mountain pass type (one positive, one negative and one sign-changing) and infinitely many high-energy sign-changing solutions.
	TO cite this article:Yi Wen, Zeng-Qi Ou. Signed and sign-changing solutions for nonlinear Schr\"{o}dinger-Poisson system with two nonlocal terms in $\mathbb{R}^{3}$[OL].[ 3 March 2023] http://en.paper.edu.cn/en_releasepaper/content/4759230


	4. Ground state sign-changing solutions for a quasilinear Schr\"{o}dinger equation
	ZHAO Yan-Ping,WU Xing-Ping,TANG Chun-Lei
	Mathematics 02 February 2023
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	Abstract:\ In this paper, we consider the existence of ground state solution and ground state sign-changing solution for the quasilinear Schr\"{o}dinger equation\begin{eqnarray}\label{1.1}-\triangle u-a(x)\triangle (u^2)u+V(x)u=f(x,u) &x\in \mathbb{R}^N\nonumber\end{eqnarray}where $N\geq3$, $V$ is coercive potential, $a(x)$ is a bounded function and $f\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$. The proof is based on variational methods, by using sign-changing Nehari manifold and deformation arguments, we can get a least energy sign-changing solution.
	TO cite this article:ZHAO Yan-Ping,WU Xing-Ping,TANG Chun-Lei. Ground state sign-changing solutions for a quasilinear Schr\"{o}dinger equation[OL].[ 2 February 2023] http://en.paper.edu.cn/en_releasepaper/content/4758883


	5. Existence of a positive solution to Kirchhoff type problemswithout compactness conditions
	DENG Landan, SHANG Yanying
	Mathematics 29 December 2021
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	Abstract:In this paper, we consider a class of Kirchhoff problem with steep potential well,the nonlinear term does not require usual compactness conditions, under appropriate assumptions, we establish the existence of positive solutions by utilizing the truncation technique to overcome the lack of compactness.
	TO cite this article:DENG Landan, SHANG Yanying. Existence of a positive solution to Kirchhoff type problemswithout compactness conditions[OL].[29 December 2021] http://en.paper.edu.cn/en_releasepaper/content/4755924


	6. Ground state solution for Schr\"{o}dinger-KdV system with asymptotically periodic potential
	Liang Fei-Fei,Wu Xing-Ping,Tang Chun-Lei
	Mathematics 08 March 2021
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	Abstract:In this paper, we study the coupled nonlinear Schr\"{o}dinger-Korteweg-de Vries system with asymptotically periodic potential. By using the variational method and Nehari manifold, we obtain the existence of non-trivial ground state solution in dimensions $N\leq3$.
	TO cite this article:Liang Fei-Fei,Wu Xing-Ping,Tang Chun-Lei. Ground state solution for Schr\"{o}dinger-KdV system with asymptotically periodic potential[OL].[ 8 March 2021] http://en.paper.edu.cn/en_releasepaper/content/4753986


	7. Infinitely many high energy radial solutions for Chern-Simons-Schr\"{o}dinger systems
	YuYanyan,Tang Chunlei
	Mathematics 03 April 2020
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	Abstract:In this paper, we investigate the following Chern-Simons-Schr\"{o}dinger system\begin{equation}\label{css}\begin{cases} -\Delta u+ u+A_{0}u+A_{1}^{2}u+A_{2}^{2}u=f(u), \\ \partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}u^{2},\qquad\partial_{1}A_{1}+\partial_{2}A_{2}=0,\\ \partial_{1}A_{0}=A_{2}u^{2}, \qquad \partial_{2}A_{0}=-A_{1}u^{2},\end{cases}\end{equation}where $\partial_{1}=\frac{\partial}{\partial x_{1}}, \partial_{2}=\frac{\partial}{\partial x_{2}}$ for $x=\left(x_{1}, x_{2}\right) \in \mathbb{R}^{2}$, $A_{j} : \mathbb{R}^{2} \rightarrow \mathbb{R}$ is the gauge field $(j=0,1,2)$. If $f$ satisfies the suitable subcritical conditions.By using variational methods, we prove that Chern-Simons-Schr\"{o}dinger system has infinitely many high energy radial solutions.
	TO cite this article:YuYanyan,Tang Chunlei. Infinitely many high energy radial solutions for Chern-Simons-Schr\"{o}dinger systems[OL].[ 3 April 2020] http://en.paper.edu.cn/en_releasepaper/content/4751383


	8. On super weak compactness of subsets and its equivalences in Banach spaces
	CHENG Li-Xin,CHENG Qing-Jin,TU Kun,ZHANG Ji-Chao
	Mathematics 26 April 2017
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	Abstract:Analogous to weak compactness of subsets of Banach spaces and to property of subsets in super reflexive spaces, the purpose of this paper is to discuss super weak compactness of both convex and nonconvex subsets in Banach spaces. As a result, this paper gives two characterizations of super weakly compact sets: The first one is Grothendiek's type theorem; the second one is James' type characterization. These are done by localizing some basic properties of ultrapowers and using some geometric procedures of Banach spaces.
	TO cite this article:CHENG Li-Xin,CHENG Qing-Jin,TU Kun, et al. On super weak compactness of subsets and its equivalences in Banach spaces[OL].[26 April 2017] http://en.paper.edu.cn/en_releasepaper/content/4729884


	9. A new approach to measure of non-compactness of Banach spaces
	CHENG Li-Xin,CHENG Qing-Jin,SHEN Qin-Rui,TU Kun,ZHANG Wen
	Mathematics 26 April 2017
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	Abstract:This paper aims to deal with measures of noncompactness of a Banach space $X$ in a new way: Assume that $mathfrak C$ is the collection of all nonempty bounded closed convex sets of $X$, $mathfrak Ksubsetmathfrak C$ consisting of all compact convex sets and $Omega$ is the closed unit ball of the dual $X^*$. Then (1); $mathfrak C$ is a normed semigroup endowed with the set addition $Aoplus B=overline{A+B}$, the usual scaler multiplication of sets and endowed with the norm $\|\|cdot\|\|$ defined for $Cinmathfrak C$ by $\|\|C\|\|=sup_{cin C}\|c\|$; (2); $J: mathfrak C ightarrow C_b(Omega)$ defined by $JC=sup_{cin C}langlecdot,c angle$ is a positively linear order isometry; further (3); both $E_mathfrak C=overline{Jmathfrak C-Jmathfrak C}$ and $E_mathfrak K=overline{Jmathfrak K-Jmathfrak K}$ are Banach sublattices and $E_mathfrak K$ is a lattice ideal of $E_mathfrak C$;(4) the quotient space $Q(E_mathfrak C)equiv E_mathfrak C/E_mathfrak K$ is an abstract $M$ space; consequently, it is order isometric to a sublattice $T(E_mathfrak C/E_mathfrak K)$ of a $C(K)$ space for some compact Hausdorff space $K$.
	TO cite this article:CHENG Li-Xin,CHENG Qing-Jin,SHEN Qin-Rui, et al. A new approach to measure of non-compactness of Banach spaces[OL].[26 April 2017] http://en.paper.edu.cn/en_releasepaper/content/4729878


	10. On the Lagrangian boundary problem of Hamiltonian systems and Seifert conjecture
	Liu Chungen
	Mathematics 07 March 2014
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	Abstract:In this survey paper, we give a brief introduction to the Lagrangian boundary problem of Hamiltonian systems,the famous Seifert conjecture and some recent progress.
	TO cite this article:Liu Chungen. On the Lagrangian boundary problem of Hamiltonian systems and Seifert conjecture[OL].[ 7 March 2014] http://en.paper.edu.cn/en_releasepaper/content/4589205

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