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    1. HILBERT PROBLEM 15 AND NONSTANDARD ANALYSIS (I)
    李邦河
    数学物理学报(英文版)    2020, 40 (1): 1-15.   DOI: 10.1007/s10473-020-0101-4
    摘要79)      PDF       收藏
    In this article, we consider the following coupled fractional nonlinear Schr?dinger system in $\mathbb{R}^{N}$ \[\left\{ \begin{array}{l} {\left( { - \Delta } \right)^s}u + P\left( x \right)u = {\mu _1}{\left| u \right|^{2p - 2}}u + \beta {\left| u \right|^p}{\left| u \right|^{p - 2}}u,\;\;\;x \in {{\mathbb{R}}^N},\\{\left( { - \Delta } \right)^s}v + Q\left( x \right)v = {\mu _2}{\left| v \right|^{2p - 2}}v + \beta {\left| v \right|^p}{\left| v \right|^{p - 2}}v,\;\;\;\;\;x \in {{\mathbb{R}}^N},\\u,\;\;v \in {H^s}\left( {{{\mathbb{R}}^N}} \right),\end{array} \right.\] where $N≥2, 0 < s < 1, 1 < p < \frac{N}{N-2s},\mu_1>0, \mu_2>0$ and $\beta \in \mathbb{R}$ is a coupling constant. We prove that it has infinitely many non-radial positive solutions under some additional conditions on $P(x), Q(x), p$ and $\beta$. More precisely, we will show that for the attractive case, it has infinitely many non-radial positive synchronized vector solutions, and for the repulsive case, infinitely many non-radial positive segregated vector solutions can be found, where we assume that $P(x)$ and $Q(x)$ satisfy some algebraic decay at infinity.
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    2. INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY $_2F_1(\tfrac12)$-SUMS
    初文昌
    数学物理学报(英文版)    2020, 40 (2): 293-315.   DOI: 10.1007/s10473-020-0201-y
    摘要49)      PDF       收藏
    The unified Ω-series of the Gauss and Bailey $_2F_1(\tfrac12)$-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts. Several remarkable transformation theorems for the Ω-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type, including a couple of beautiful expressions for π and the Catalan constant discovered by Guillera (2008).
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    3. ON THE DISTRIBUTION OF JULIA SETS OF HOLOMORPHIC MAPS
    曹春雷, 王跃飞
    数学物理学报(英文版)    2020, 40 (4): 903-909.   DOI: 10.1007/s10473-020-0401-5
    摘要49)      PDF       收藏
    In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines. Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of ${\bf C^*}$ can neither contain a free Jordan arc nor be contained in any finite set of straight lines.
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    4. INFINITELY MANY SOLITARY WAVES DUE TO THE SECOND-HARMONIC GENERATION IN QUADRATIC MEDIA
    王春花, 周静
    数学物理学报(英文版)    2020, 40 (1): 16-34.   DOI: 10.1007/s10473-020-0102-3
    摘要45)      PDF       收藏

    In this paper, we consider the following coupled Schr?dinger system with χ(2) nonlinearities

    which arises from second-harmonic generation in quadratic media. Here V1(x) and V2(x) are radially positive functions, 2 ≤ N < 6, α > 0 and α > β. Assume that the potential functions V1(x) and V2(x) satisfy some algebraic decay at infinity. Applying the finite dimensional reduction method, we construct an unbounded sequence of non-radial vector solutions of synchronized type.

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    5. MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY
    郭伦, 胡亭曦
    数学物理学报(英文版)    2020, 40 (2): 316-340.   DOI: 10.1007/s10473-020-0202-x
    摘要43)      PDF       收藏
    In this article, we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity \begin{equation*} -\Delta u+(\lambda a(x)+1)u=\Big(\frac{1}{|x|^{\alpha}}\ast F(u)\Big)f(u) \ \ \text{in}\ \ \mathbb{R}^{N}, \end{equation*} where $N\geq 3$, $0<\alpha< \min\{N,4\}$, $\lambda$ is a positive parameter and the nonnegative potential function $a(x)$ is continuous. Using variational methods, we prove that if the potential well int$(a^{-1}(0))$ consists of $k$ disjoint components, then there exist at least $2^k-1$ multi-bump solutions. The asymptotic behavior of these solutions is also analyzed as $\lambda\to +\infty$.
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    6. COMPLEX SYMMETRIC TOEPLITZ OPERATORS ON THE UNIT POLYDISK AND THE UNIT BALL
    蒋操, 董兴堂, 周泽华
    数学物理学报(英文版)    2020, 40 (1): 35-44.   DOI: 10.1007/s10473-020-103-2
    摘要39)      PDF       收藏
    In this article, we study complex symmetric Toeplitz operators on the Bergman space and the pluriharmonic Bergman space in several variables. Surprisingly, the necessary and sufficient conditions for Toeplitz operators to be complex symmetric on these two spaces with certain conjugations are just the same. Also, some interesting symmetry properties of complex symmetric Toeplitz operators are obtained.
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    7. THE BOUNDEDNESS FOR COMMUTATORS OF ANISOTROPIC CALDERÓN-ZYGMUND OPERATORS
    李金霞, 李宝德, 何建勋
    数学物理学报(英文版)    2020, 40 (1): 45-58.   DOI: 10.1007/s10473-020-104-1
    摘要33)      PDF       收藏
    Let T be an anisotropic Calderón-Zygmund operator and φ : Rn×[0, ∞) → [0, ∞) be an anisotropic Musielak-Orlicz function with φ(x, ·) being an Orlicz function and φ(·, t) being a Muckenhoupt A(A) weight. In this paper, our goal is to study two boundedness theorems for commutators of anisotropic Calderón-Zygmund operators. Precisely, when b ∈ BMOw(Rn, A) (a proper subspace of anisotropic bounded mean oscillation space BMO(Rn, A)), the commutator [b, T] is bounded from anisotropic weighted Hardy space Hw1(Rn, A) to weighted Lebesgue space Lw1(Rn) and when b ∈ BMO(Rn) (bounded mean oscillation space), the commutator [b, T] is bounded on Musielak-Orlicz space Lφ(Rn), which are extensions of the isotropic setting.
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    8. TOEPLITZ OPERATORS WITH POSITIVE OPERATOR-VALUED SYMBOLS ON VECTOR-VALUED GENERALIZED FOCK SPACES
    陈建军, 王晓峰, 夏锦
    数学物理学报(英文版)    2020, 40 (3): 625-640.   DOI: 10.1007/s10473-020-0303-6
    摘要30)      PDF       收藏
    In this article, we study some characterizations of Toeplitz operators with positive operator-valued function as symbols on the vector-valued generalized Bargmann-Fock spaces Fψ2. Main results including Fock-Carleson condition, bounded Toeplitz operators, compact Toeplitz operators, and Toeplitz operators in the Schatten-p class are all considered.
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    9. A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES
    沈钦锐
    数学物理学报(英文版)    2020, 40 (3): 603-613.   DOI: 10.1007/s10473-020-0301-8
    摘要29)      PDF       收藏
    This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection C(X) (consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication) is a normed semigroup, and the mapping J from C(X) onto F(Ω) is a fully order-preserving positively linear surjective isometry, where Ω is the closed unit ball of X* and F(Ω) the collection of all continuous and w*-lower semicontinuous sublinear functions on X* but restricted to Ω. Furthermore, both EC=JC-JC and EK=JK-JK are Banach lattices and EK is a lattice ideal of EC. The quotient space EC/EK is an abstract M space, hence, order isometric to a sublattice of C(K) for some compact Haudorspace K, and (FQJ)C which is a closed cone is contained in the positive cone of C(K), where Q:ECEC/EK is the quotient mapping and F:EC/EKC(K) is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given:Let BX be the closed unit ball of a Banach space X, then
    μ(T)=μ(T(BX))=||(F QJ)T(BX)||C(K), ∀TB(X).
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    10. ASYMPTOTIC BEHAVIOR OF SOLUTION BRANCHES OF NONLOCAL BOUNDARY VALUE PROBLEMS
    徐西安, 秦宝侠, 王震
    数学物理学报(英文版)    2020, 40 (2): 341-354.   DOI: 10.1007/s10473-020-0203-9
    摘要28)      PDF       收藏
    In this article, by employing an oscillatory condition on the nonlinear term, a result is proved for the existence of connected component of solutions set of a nonlocal boundary value problem, which bifurcates from infinity and asymptotically oscillates over an interval of parameter values. An interesting and immediate consequence of such oscillation property of the connected component is the existence of infinitely many solutions of the nonlinear problem for all parameter values in that interval.
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    11. BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS
    李宗广, 刘锐
    数学物理学报(英文版)    2020, 40 (4): 910-920.   DOI: 10.1007/s10473-020-0402-4
    摘要28)      PDF       收藏
    In this article, we study the blow-up solutions for a case of b-family equations. Using the qualitative theory of differential equations and the bifurcation method of dynamical systems, we obtain five types of blow-up solutions: the hyperbolic blow-up solution, the fractional blow-up solution, the trigonometric blow-up solution, the first elliptic blow-up solution, and the second elliptic blow-up solution. Not only are the expressions of these blow-up solutions given, but also their relationships are discovered. In particular, it is found that two bounded solitary solutions are bifurcated from an elliptic blow-up solution.
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    12. BOUNDEDNESS OF MULTILINEAR LITTLEWOOD-PALEY OPERATORS ON AMALGAM-CAMPANATO SPACES
    李翔, 骞君, 燕敦验
    数学物理学报(英文版)    2020, 40 (1): 272-292.   DOI: 10.1007/s10473-020-0119-7
    摘要27)      PDF       收藏
    In this paper, we consider the boundedness of multilinear Littlewood-Paley operators which include multilinear g-function, multilinear Lusin’s area integral and multilinear Littlewood-Paley gλ*-function. Furthermore, norm inequalities of the above operators hold on the corresponding Amalgam-Campanato spaces.
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    13. ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRÖDINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS
    Xuan Truong LE, Thanh Nhan NGUYEN, Ngoc Trong NGUYEN
    数学物理学报(英文版)    2020, 40 (5): 1171-1184.   DOI: 10.1007/s10473-020-0501-2
    摘要26)      PDF       收藏
    In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrödinger operator ?=-△+V in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Hölder's inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrödinger equations in the new Morrey space.
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    14. GROUND STATES FOR FRACTIONAL SCHRÖDINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS AND CRITICAL GROWTH
    李全清, 王文波, 滕凯民, 吴鲜
    数学物理学报(英文版)    2020, 40 (1): 59-74.   DOI: 10.1007/s10473-020-0105-0
    摘要22)      PDF       收藏
    In this article, we study the following fractional Schrödinger equation with electromagnetic fields and critical growth
    (-△)Asu + V (x)u = |u|2s*-2u + λf(x,|u|2)u, x ∈ RN,
    where (-△)As is the fractional magnetic operator with 0 < s < 1, N > 2s, λ > 0, 2s* = 2N/(N-2s), f is a continuous function, VC(RN, R) and AC(RN, RN) are the electric and magnetic potentials, respectively. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by Nehari method.
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    15. ON THE EXISTENCE OF SOLUTIONS TO A BI-PLANAR MONGE-AMPÈRE EQUATION
    Ibrokhimbek AKRAMOV, Marcel OLIVER
    数学物理学报(英文版)    2020, 40 (2): 379-388.   DOI: 10.1007/s10473-020-0206-6
    摘要21)      PDF       收藏
    In this article, we consider a fully nonlinear partial differential equation which can be expressed as a sum of two Monge-Ampère operators acting in different two-dimensional coordinate sections. This equation is elliptic, for example, in the class of convex functions. We show that the notion of Monge-Ampère measures and Aleksandrov generalized solutions extends to this equation, subject to a weaker notion of convexity which we call bi-planar convexity. While the equation is also elliptic in the class of bi-planar convex functions, the contrary is not necessarily true. This is a substantial difference compared to the classical Monge-Ampère equation where ellipticity and convexity coincide. We provide explicit counter-examples: classical solutions to the bi-planar equation that satisfy the ellipticity condition but are not generalized solutions in the sense introduced. We conclude that the concept of generalized solutions based on convexity arguments is not a natural setting for the bi-planar equation.
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    16. THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK POSITIVE SOLUTIONS TO A CLASS OF KIRCHHOFF EQUATION
    李工宝, 牛亚慧
    数学物理学报(英文版)    2020, 40 (1): 90-112.   DOI: 10.1007/s10473-020-0107-y
    摘要19)      PDF       收藏
    In the present paper, we consider the nonlocal Kirchhoff problem
    - (ε2a + εbR3 |▽u|2)△u + u = Q(x)up, u > 0 in R3,
    where a, b > 0, 1 < p < 5 and ε > 0 is a parameter. Under some assumptions on Q(x), we show the existence and local uniqueness of positive multi-peak solutions by LyapunovSchmidt reduction method and the local Pohozaev identity method, respectly.
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    17. MINIMAL PERIOD SYMMETRIC SOLUTIONS FOR SOME HAMILTONIAN SYSTEMS VIA THE NEHARI MANIFOLD METHOD
    Chouha?d SOUISSI
    数学物理学报(英文版)    2020, 40 (3): 614-624.   DOI: 10.1007/s10473-020-0302-7
    摘要18)      PDF       收藏
    For a given T > 0, we prove, under the global ARS-condition and using the Nehari manifold method, the existence of a T-periodic solution having the W-symmetry introduced in[21], for the hamiltonian system
    z+ V'(z)=0, z ∈ RN, N ∈ N*.
    Moreover, such a solution is shown to have T as a minimal period without relaying to any index theory. A multiplicity result is also proved under the same condition.
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    18. LOCAL WELL-POSEDNESS OF STRONG SOLUTIONS FOR THE NONHOMOGENEOUS MHD EQUATIONS WITH A SLIP BOUNDARY CONDITIONS
    李红民, 肖跃龙
    数学物理学报(英文版)    2020, 40 (2): 442-456.   DOI: 10.1007/s10473-020-0210-x
    摘要18)      PDF       收藏
    This article is concerned with the 3D nonhomogeneous incompressible magnetohydrodynamics equations with a slip boundary conditions in bounded domain. We obtain weighted estimates of the velocity and magnetic field, and address the issue of local existence and uniqueness of strong solutions with the weaker initial data which contains vacuum states.
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    19. UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION
    赵文强, 张一进
    数学物理学报(英文版)    2020, 40 (4): 921-933.   DOI: 10.1007/s10473-020-0403-3
    摘要18)      PDF       收藏
    In this paper, we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit. As an application, we obtain the convergence of random attractors for non-autonomous stochastic reaction-diffusion equations on unbounded domains, when the density of stochastic noises approaches zero. The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem. A differentiability condition on nonlinearity is omitted, which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity. These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.
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    20. GROUND STATE SOLUTIONS FOR A SCHRÖDINGER-POISSON SYSTEM WITH UNCONVENTIONAL POTENTIAL
    杜瑶, 唐春雷
    数学物理学报(英文版)    2020, 40 (4): 934-944.   DOI: 10.1007/s10473-020-0404-2
    摘要18)      PDF       收藏
    We consider the Schrödinger-Poisson system with nonlinear term $Q(x)|u|^{p-1}u$, where the value of $\displaystyle\lim_{|x|\rightarrow\infty} Q(x)$ may not exist and $Q$ may change sign. This means that the problem may have no limit problem. The existence of nonnegative ground state solutions is established. Our method relies upon the variational method and some analysis tricks.
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