中国科学院数学与系统科学研究院期刊网
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15 August 2025, Volume 41 Issue 8
    

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  • The Ground State Solutions for the Choquard Equation with p-Laplacian on Finite Lattice Graphs
    Yang Liu, Mengjie Zhang
    Acta Mathematica Sinica. 2025, 41(8): 1953-1965. https://doi.org/10.1007/s10114-025-3223-5
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    In this paper, we study the $p$-Laplacian Choquard equation $$-△_p u+V(x)|u|^{p-2}u=\bigg({\sum_{y\in N^n\atop y\not=x}}\frac{|u(y)|^q}{d(x,\,y)^{n-\alpha}}\bigg)|u|^{q-2}u$$ on a finite lattice graph $N^n$ with $n\in\mathbb{N}_+$, where $p>1,$ $q>1$ and $0\leq\alpha\leq n$ are some constants, $V(x)$ is a positive function on $N^n$. Using the Nehari method, we prove that if 1<p<q<+∞, then the above equation admits a ground state solution. Previously, the $p$-Laplacian Choquard equation on finite lattice graph has not been studied, and our result contains the critical cases $\alpha=0$ and $\alpha=n$, which further improves the study of Choquard equations on lattice graphs.
  • Relative Regionally Proximal Tuples and Sensitivity
    Yini Yang
    Acta Mathematica Sinica. 2025, 41(8): 1966-1976. https://doi.org/10.1007/s10114-025-3392-2
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    First we investigate relative $n$-regionally proximal tuples. Let $\pi: (X,G)→ (Y,G)$ be a Bronstein extension between minimal systems. It turns out that if $(x_1,\dots, x_n)$ is a minimal point and $(x_{i},x_{i+1})$ is relative regionally proximal for $1\leq i\leq n-1$, then $(x_1,\dots, x_n)$ is relative $n$-regionally proximal. We consider the relative versions of sensitivity, including relative $n$-sensitivity and relative block $F_t$-$n$-sensitivity, where $F_t$ is the family of thick sets. We show that $\pi$ is relatively $n$-sensitive if and only if the relative $n$-regionally proximal relation contains a point whose coordinates are distinct, and the structure of $\pi$ which is relatively $n$-sensitive but not relatively $n+1$-sensitive is determined. We also characterize relatively block $F_t$-$n$-sensitive via relative regionally proximal tuples.
  • Quasilinear Elliptic Problems with Exponential Growth via the Nehari Manifold Method: Existence of Nonnegative and Nodal Solutions
    Giovany Figueiredo, Sandra Moreira, Ricardo Ruviaro
    Acta Mathematica Sinica. 2025, 41(8): 1977-1994. https://doi.org/10.1007/s10114-025-4053-1
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    In this paper we will be concerned with the problem $$ - \Delta u - \frac{1}{2}\Delta(a(x)u^2) u + V(x)u=f(u), x\in \mathbb{R}^2, $$ where $V$ is a potential continuous and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a superlinear continuous function with exponential subcritical or exponential critical growth. We use as a main tool the Nehari manifold method in order to show existence of nonnegative solutions and existence of nodal solutions. Our results complement the classical result of Solutions for quasilinear Schr?dinger equations via the Nehari method" due to Jia-Quan Liu, Ya-Qi Wang and Zhi-Qiang Wang in the sense that in this article we are considering nonlinearity of the exponential type.
  • Complete Kähler Metrics with Positive Holomorphic Sectional Curvatures on Certain Line Bundles (Related to a Co-Homogeneity One Point of View on an Yau Conjecture) II
    Daniel Guan, Mengxiang Liang
    Acta Mathematica Sinica. 2025, 41(8): 1995-2010. https://doi.org/10.1007/s10114-025-3371-7
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    In this article, we continue to study K?hler metrics on line bundles over projective spaces to find complete K?hler metrics with positive holomorphic sectional curvatures with two very special properties. These two special kinds of examples were not able to be found in our earlier paper of the first author and Ms. Duan. And therefore, we give a further step toward a famous Yau conjecture with the method in the co-homogeneity one geometry.
  • Homoclinic Solutions for a Class of Asymptotically Quadratic Second-Order Hamiltonian System
    Tiefeng Ye, Huixing Zhang, Wenbin Liu
    Acta Mathematica Sinica. 2025, 41(8): 2011-2030. https://doi.org/10.1007/s10114-025-4041-5
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    In this paper, we study the existence and multiplicity of homoclinic solutions for a class of second-order Hamiltonian system: $u''(t)-L(t)u(t) + \nabla V(t,u) = 0$, where $L(t)$ and $V(t,u)$ are not periodic in $t$. First, we introduce the definition of index and establish the corresponding index theory. Then, by using the index theory and critical point theory, we prove our main results under the asymptotic quadratic conditions of the potential function.
  • Existence of Normalized Ground State Solutions for Quasilinear Elliptic Problems in RN
    Yuanyuan Li, Jingbo Dou
    Acta Mathematica Sinica. 2025, 41(8): 2031-2052. https://doi.org/10.1007/s10114-025-3547-1
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    In this paper, we investigate the existence of normalized solutions for a quasilinear elliptic problem as follows \begin{equation*} \left\{\begin{array}{ll} -\Delta_p u+\lambda u^{p-1}=f(u), & x\in \mathbb{R}^N, \\ \displaystyle\int_{\mathbb{R}^N}|u|^p d x=\rho,& u \in W^{1,p}(\mathbb{R}^N), \end{array}\right. \end{equation*} where $-\Delta_p $ is the $p$-Laplace operator, 1<p<N,N≥3,ρ>0 and λ>0. f is a continuous function and satisfies some suitable conditions. Based on a Nehari—Pohozaev manifold, we show the existence of positive normalized solutions by using the minimization method.
  • The Mean Orbital Pseudo-metric and the Space of Invariant Measures
    Jian Li, Yuanfen Xiao
    Acta Mathematica Sinica. 2025, 41(8): 2053-2071. https://doi.org/10.1007/s10114-025-3168-8
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    We study the mean orbital pseudo-metric for Polish dynamical systems and its connections with properties of the space of invariant measures. We give equivalent conditions for when the set of invariant measures generated by periodic points is dense in the set of ergodic measures and the space of invariant measures. We also introduce the concept of asymptotic orbital average shadowing property and show that it implies that every non-empty compact connected subset of the space of invariant measures has a generic point.
  • Perturbations of Dirac Operators, Spectral Einstein Functionals and the Noncommutative Residue
    Sining Wei, Yong Wang
    Acta Mathematica Sinica. 2025, 41(8): 2072-2104. https://doi.org/10.1007/s10114-025-3654-z
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    In this paper, we introduce the spectral Einstein functional for perturbations of Dirac operators on manifolds with boundary. Furthermore, we provide the proof of the Dabrowski—Sitarz—Zalecki type theorems associated with the spectral Einstein functionals for perturbations of Dirac operators, particularly in the cases of on 4-dimensional manifolds with boundary.
  • Observability for the Schrödinger Equation in a Uniform Magnetic Field
    Fang Zhang
    Acta Mathematica Sinica. 2025, 41(8): 2105-2127. https://doi.org/10.1007/s10114-025-4389-6
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    We prove the observability inequalities at two time points for the Schr?dinger equation in a uniform magnetic field in dimensions 2 and 3. The proofs mainly rely on Nazarov's uncertainty principle. In particular, the observability inequality in three dimensions can also be derived from the approach used to establish the Amerin—Berthier uncertainty principle.
  • On Lu Qi-Keng Uniformization Theorem for Stein Spaces with Singularities
    Zhenqian Li, Zhi Li
    Acta Mathematica Sinica. 2025, 41(8): 2128-2138. https://doi.org/10.1007/s10114-025-4447-0
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    In this article, we show that the universal covering of any complete normal K?hler space of constant holomorphic sectional curvature on the regular locus is exactly biholomorphic to one of the complex projective space, the complex Euclidean space or the complex Euclidean ball. Moreover, we also prove that in a normal Stein space any bounded domain with complete Bergman metric of constant holomorphic sectional curvature on the regular locus is necessarily biholomorphic to the complex Euclidean ball, by which we generalize the classical Lu Qi-Keng uniformization theorem to the singular setting.
  • Large Deviation Rates for Supercritical Multitype Branching Processes with Immigration
    Liuyan Li, Junping Li
    Acta Mathematica Sinica. 2025, 41(8): 2139-2159. https://doi.org/10.1007/s10114-025-3051-7
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    Let $\{X_n\}_{n\geq0}$ be a $p$-type ($p\geq2$) supercritical branching process with immigration and mean matrix $M$. Suppose that $M$ is positively regular and $\rho$ is the maximal eigenvalue of $M$ with the corresponding left and right eigenvectors $\boldsymbol{v}$ and $\boldsymbol{u}$. Let $\rho>1$ and $Y_n=\rho^{-n}[\boldsymbol{u}\cdot X_n -\frac{\rho^{n+1}-1}{\rho-1}( \boldsymbol{u}\cdot \boldsymbol{\lambda})]$, where the vector $\boldsymbol{\lambda}$ denotes the mean immigration rate. In this paper, we will show that $Y_n$ is a martingale and converges to an $r.v.$ $Y$ as $n\rightarrow\infty$. We study the rates of convergence to $0$ as $n\rightarrow\infty$ of $$ P_i\bigg(\bigg|\frac{\boldsymbol{l}\cdot X_{n+1}}{\textbf{1}\cdot X_n}-\frac{\boldsymbol{l}\cdot(X_nM)}{\textbf{1}\cdot X_n}\bigg|>\varepsilon\bigg), P_i\bigg(\bigg|\frac{\boldsymbol{l}\cdot X_n}{\textbf{1}\cdot X_n}-\frac{\boldsymbol{l}\cdot\boldsymbol{v}}{\textbf{1}\cdot \boldsymbol{v} }\bigg|>\varepsilon\bigg), P(|Y_n-Y|>\varepsilon) $$ for any $\varepsilon>0,\, i=1,\dots,p$, $\textbf{1}=(1,\dots,1)$ and $\boldsymbol{l}\in\mathbb{R}^p,$ the $p$-dimensional Euclidean space. It is shown that under certain moment conditions, the first two decay geometrically, while conditionally on the event $Y\geq\alpha\ (\alpha>0)$ supergeometrically. The decay rate of the last probability is always supergeometric under a finite moment generating function assumption.
  • A Half-Proximal Symmetric Splitting Method for Non-Convex Separable Optimization
    Pengjie Liu, Jinbao Jian, Hu Shao, Xiaoquan Wang, Xiangfeng Wang
    Acta Mathematica Sinica. 2025, 41(8): 2160-2194. https://doi.org/10.1007/s10114-025-4144-z
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    In this paper, we explore the convergence and convergence rate results for a new methodology termed the half-proximal symmetric splitting method (HPSSM). This method is designed to address linearly constrained two-block non-convex separable optimization problem. It integrates a half-proximal term within its first subproblem to cancel out complicated terms in applications where the subproblem is not easy to solve or lacks a simple closed-form solution. To further enhance adaptability in selecting relaxation factor thresholds during the two Lagrange multiplier update steps, we strategically incorporate a relaxation factor as a disturbance parameter within the iterative process of the second subproblem. Building on several foundational assumptions, we establish the subsequential convergence, global convergence, and iteration complexity of HPSSM. Assuming the presence of the Kurdyka-?ojasiewicz inequality of ?ojasiewicz-type within the augmented Lagrangian function (ALF), we derive the convergence rates for both the ALF sequence and the iterative sequence. To substantiate the effectiveness of HPSSM, sufficient numerical experiments are conducted. Moreover, expanding upon the two-block iterative scheme, we present the theoretical results for the symmetric splitting method when applied to a three-block case.
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