#### Table of Content

25 June 2022, Volume 42 Issue 3
Articles
 BOUNDEDNESS AND EXPONENTIAL STABILIZATION IN A PARABOLIC-ELLIPTIC KELLER–SEGEL MODEL WITH SIGNAL-DEPENDENT MOTILITIES FOR LOCAL SENSING CHEMOTAXIS Jie JIANG Acta mathematica scientia,Series B. 2022, 42 (3):  825-846.  DOI: 10.1007/s10473-022-0301-y In this paper we consider the initial Neumann boundary value problem for a degenerate Keller—Segel model which features a signal-dependent non-increasing motility function. The main obstacle of analysis comes from the possible degeneracy when the signal concentration becomes unbounded. In the current work, we are interested in the boundedness and exponential stability of the classical solution in higher dimensions. With the aid of a Lyapunov functional and a delicate Alikakos—Moser type iteration, we are able to establish a time-independent upper bound of the concentration provided that the motility function decreases algebraically. Then we further prove the uniform-in-time boundedness of the solution by constructing an estimation involving a weighted energy. Finally, thanks to the Lyapunov functional again, we prove the exponential stabilization toward the spatially homogeneous steady states. Our boundedness result improves those in [1] and the exponential stabilization is obtained for the first time.
 ABSOLUTE MONOTONICITY INVOLVING THE COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND WITH APPLICATIONS Zhenhang YANG, Jingfeng TIAN Acta mathematica scientia,Series B. 2022, 42 (3):  847-864.  DOI: 10.1007/s10473-022-0302-x Let $\mathcal{K}\left( r\right)$ be the complete elliptic integrals of the first kind for $r\in \left( 0,1\right)$ and $f_{p}\left( x\right) =\left[ \left( 1-x\right) ^{p}\mathcal{K}\left( \sqrt{x}\right) \right]$. Using the recurrence method, we find the necessary and sufficient conditions for the functions $-f_{p}^{\prime }$, $\ln f_{p}$, $-\left( \ln f_{p}\right) ^{\left( i\right) }$ ($i=1,2,3$) to be absolutely monotonic on $\left( 0,1\right)$. As applications, we establish some new bounds for the ratios and the product of two complete integrals of the first kind, including the double inequalities \begin{eqnarray*} \frac{\exp \left[ r^{2}\left( 1-r^{2}\right) /64\right] }{\left( 1+r\right) ^{1/4}} &<&\frac{\mathcal{K}\left( r\right) }{\mathcal{K}\left( \sqrt{r}% \right) }<\exp \left[ -\frac{r\left( 1-r\right) }{4}\right] , \\ \frac{\pi }{2}\exp \left[ \theta _{0}\left( 1-2r^{2}\right) \right] &<&\frac{% \pi }{2}\frac{\mathcal{K}\left( r^{\prime }\right) }{\mathcal{K}\left( r\right) }<\frac{\pi }{2}\left( \frac{r^{\prime }}{r}\right) ^{p}\exp \left[ \theta _{p}\left( 1-2r^{2}\right) \right] , \\ \mathcal{K}^{2}\left( \frac{1}{\sqrt{2}}\right) &\leq &\mathcal{K}\left( r\right) \mathcal{K}\left( r^{\prime }\right) \leq \frac{1}{\sqrt{% 2rr^{\prime }}}\mathcal{K}^{2}\left( \frac{1}{\sqrt{2}}\right) \end{eqnarray*}% for $r\in \left( 0,1\right)$ and $p\geq 13/32$, where $r^{\prime }=% \sqrt{1-r^{2}}$ and $\theta _{p}=2\Gamma \left( 3/4\right) ^{4}/\pi ^{2}-p$.
 THE $\partial\bar{\partial}$-LEMMA UNDER SURJECTIVE MAPS Lingxu MENG Acta mathematica scientia,Series B. 2022, 42 (3):  865-875.  DOI: 10.1007/s10473-022-0303-9 We consider the $\partial\bar{\partial}$-lemma for complex manifolds under surjective holomorphic maps. Furthermore, using Deligne-Griffiths-Morgan-Sullivan's theorem, we prove that a product compact complex manifold satisfies the $\partial\bar{\partial}$-lemma if and only if all of its components do as well.
 PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS Meiqi LIU, Huijie QIAO Acta mathematica scientia,Series B. 2022, 42 (3):  876-886.  DOI: 10.1007/s10473-022-0304-8 This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters. First, we prove the existence and uniqueness of these equations under non-Lipschitz conditions. Second, we construct maximum likelihood estimators of these parameters and then discuss their strong consistency. Third, a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered. Finally, we estimate the errors between solutions of these equations and that of their numerical equations.
 GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW Ning-An LAI, Wei XIANG, Yi ZHOU Acta mathematica scientia,Series B. 2022, 42 (3):  887-902.  DOI: 10.1007/s10473-022-0305-7 In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. A non-existence result is established for the fan-shaped wave structure solution, including two shocks and one contact discontinuity which is a perturbation of plane waves. Therefore, unlike in the one-dimensional case, the multi-dimensional plane shocks are not stable globally. Moreover, a sharp lifespan estimate is established which is the same as the lifespan estimate for the nonlinear wave equations in both two and three space dimensions.
 ESTIMATES FOR EXTREMAL VALUES FOR A CRITICAL FRACTIONAL EQUATION WITH CONCAVE-CONVEX NONLINEARITIES Jianghao HAO, Yajing ZHANG Acta mathematica scientia,Series B. 2022, 42 (3):  903-918.  DOI: 10.1007/s10473-022-0306-6 In this paper we study the critical fractional equation with a parameter λ and establish uniform lower bounds for Λ, which is the supremum of the set of λ, related to the existence of positive solutions of the critical fractional equation.
 THE SYSTEMS WITH ALMOST BANACH-MEAN EQUICONTINUITY FOR ABELIAN GROUP ACTIONS Bin ZHU, Xiaojun HUANG, Yuan LIAN Acta mathematica scientia,Series B. 2022, 42 (3):  919-940.  DOI: 10.1007/s10473-022-0307-5 In this paper, we present the concept of Banach-mean equicontinuity and prove that the Banach-, Weyl- and Besicovitch-mean equicontinuities of a dynamic system of Abelian group action are equivalent. Furthermore, we obtain that the topological entropy of a transitive, almost Banach-mean equicontinuous dynamical system of Abelian group action is zero. As an application of our main result, we show that the topological entropy of the Banach-mean equicontinuous system under the action of an Abelian groups is zero.
 A NEW SUFFICIENT CONDITION FOR SPARSE RECOVERY WITH MULTIPLE ORTHOGONAL LEAST SQUARES Haifeng LI, Jing ZHANG Acta mathematica scientia,Series B. 2022, 42 (3):  941-956.  DOI: 10.1007/s10473-022-0308-4 A greedy algorithm used for the recovery of sparse signals, multiple orthogonal least squares (MOLS) have recently attracted quite a big of attention. In this paper, we consider the number of iterations required for the MOLS algorithm for recovery of a $K$-sparse signal $\mathbf{x}\in\mathbb{R}^n$. We show that MOLS provides stable reconstruction of all $K$-sparse signals $\mathbf{x}$ from $\mathbf{y}=\mathbf{A}\mathbf{x}+\mathbf{w}$ in $\lceil\frac{6K}{M}\rceil$ iterations when the matrix $\mathbf{A}$ satisfies the restricted isometry property (RIP) with isometry constant $\delta_{7K}\leq0.094$. Compared with the existing results, our sufficient condition is not related to the sparsity level $K$.
 CENTRAL LIMIT THEOREM AND CONVERGENCE RATES FOR A SUPERCRITICAL BRANCHING PROCESS WITH IMMIGRATION IN A RANDOM ENVIRONMENT Yingqiu LI, Xulan HUANG, Zhaohui PENG Acta mathematica scientia,Series B. 2022, 42 (3):  957-974.  DOI: 10.1007/s10473-022-0309-3 We are interested in the convergence rates of the submartingale ${W}_{n} =\frac{Z_{n}}{\Pi_{n}}$ to its limit ${W},$ where $(\Pi_{n})$ is the usually used norming sequence and $(Z_{n})$ is a supercritical branching process with immigration $(Y_{n})$ in a stationary and ergodic environment $\xi$. Under suitable conditions, we establish the following central limit theorems and results about the rates of convergence in probability or in law: (i) $W-W_{n}$ with suitable normalization converges to the normal law $N(0,1)$, and similar results also hold for $W_{n+k}-W_{n}$ for each fixed $k\in \mathbb{N}^{\ast};$ (ii) for a branching process with immigration in a finite state random environment, if $W_{1}$ has a finite exponential moment, then so does $W,$ and the decay rate of $\mathbb{P}(|W-W_{n}|>\varepsilon)$ is supergeometric; (iii) there are normalizing constants $a_{n}(\xi)$ (that we calculate explicitly) such that $a_{n}(\xi)(W-W_{n})$ converges in law to a mixture of the Gaussian law.
 QUASILINEAR EQUATIONS USING A LINKING STRUCTURE WITH CRITICAL NONLINEARITIES Edcarlos D. SILVA, Jefferson S. SILVA Acta mathematica scientia,Series B. 2022, 42 (3):  975-1002.  DOI: 10.1007/s10473-022-0310-x It is to establish existence of a weak solution for quasilinear elliptic problems assuming that the nonlinear term is critical. The potential V is bounded from below and above by positive constants. Because we are considering a critical term which interacts with higher eigenvalues for the linear problem, we need to apply a linking theorem. Notice that the lack of compactness, which comes from critical problems and the fact that we are working in the whole space, are some obstacles for us to ensure existence of solutions for quasilinear elliptic problems. The main feature in this article is to restore some compact results which are essential in variational methods. Recall that compactness conditions such as the Palais-Smale condition for the associated energy functional is not available in our setting. This difficulty is overcame by taking into account some fine estimates on the critical level for an auxiliary energy functional.
 SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS Nacira AGRAM, Saloua LABED, Bernt ØKSENDAL, Samia YAKHLEF Acta mathematica scientia,Series B. 2022, 42 (3):  1003-1017.  DOI: 10.1007/s10473-022-0311-9 This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations, where the solution $X^{u,\xi}(t)=X(t)$ is given by $$X(t) =\phi(t)+{ \int_{0}^{t}}b\left( t,s,X(s),u(s)\right){\rm d}s+% { \int_{0}^{t}} \sigma\left( t,s,X(s),u(s)\right) {\rm d}B(s) + { \int_{0}^{t}} h\left( t,s\right) {\rm d}\xi(s).$$ Here ${\rm d}B(s)$ denotes the Brownian motion Itô type differential, $\xi$ denotes the singular control (singular in time $t$ with respect to Lebesgue measure) and $u$ denotes the regular control (absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory, where $X(t)$ represents the population density at time $t$, and the singular control process $\xi$ represents the harvesting effort rate. The total income from the harvesting is represented by $$J(u,\xi) =\mathbb{E}\bigg[ \int_{0}^{T} f_{0}(t,X(t),u(t)){\rm d}t+ \int_{0}^{T} f_{1}(t,X(t)){\rm d}\xi(t)+g(X(T))\bigg]$$ for the given functions $f_{0},f_{1}$ and $g$, where $T>0$ is a constant denoting the terminal time of the harvesting. Note that it is important to allow the controls to be singular, because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus, we prove sufficient conditions and necessary conditions of optimality of controls. As a consequence, we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally, to illustrate our results, we apply them to discuss optimal harvesting problems with possibly density dependent prices.
 HE EXPONENTIAL OF QUASI BLOCK-TOEPLITZ MATRICES Elahe BOLOURCHIAN, Bijan Ahmadi KAKAVANDI Acta mathematica scientia,Series B. 2022, 42 (3):  1018-1034.  DOI: 10.1007/s10473-022-0312-8 The matrix Wiener algebra, $\mathcal{W}_N:=\mathrm{M}_{N} (\mathcal{W})$ of order $N>0$, is the matrix algebra formed by $N \times N$ matrices whose entries belong to the classical Wiener algebra $\mathcal{W}$ of functions with absolutely convergent Fourier series. A block-Toeplitz matrix $T(a)=[A_{i,j}]_{i,j \geq 0}$ is a block semi-infinite matrix such that its blocks $A_{i,j}$ are finite matrices of order $N$, $A_{i,j}=A_{r,s}$ whenever $i-j=r-s$ and its entries are the coefficients of the Fourier expansion of the generator $a:\mathbb{T} \rightarrow \mathrm{M}_{N} (\mathbb{C})$. Such a matrix can be regarded as a bounded linear operator acting on the direct sum of $N$ copies of $L^2(\mathbb{T})$. We show that $\exp(T(a))$ differes from $T(\exp(a))$ only in a compact operator with a known bound on its norm. In fact, we prove a slightly more general result: for every entire function $f$ and for every compact operator $E$, there exists a compact operator $F$ such that $f(T(a)+E)=T(f(a))+F$. We call these $T(a)+E's$ matrices, the quasi block-Toeplitz matrices, and we show that via a computation-friendly norm, they form a Banach algebra. Our results generalize and are motivated by some recent results of Dario Andrea Bini, Stefano Massei and Beatrice Meini.
 BOUNDEDNESS AND ASYMPTOTIC STABILITY IN A PREDATOR-PREY CHEMOTAXIS SYSTEM WITH INDIRECT PURSUIT-EVASION DYNAMICS Shuyan QIU, Chunlai MU, Hong YI Acta mathematica scientia,Series B. 2022, 42 (3):  1035-1057.  DOI: 10.1007/s10473-022-0313-7 This work explores the predator-prey chemotaxis system with two chemicals \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\Delta u+\chi\nabla\cdot(u\nabla v)+\mu_1u(1-u-a_1w),\quad &x\in \Omega, t>0,\\ v_t=\Delta v-\alpha_1 v+\beta_1w,\quad &x\in \Omega, t>0,\\ w_t=\Delta w-\xi\nabla \cdot(w\nabla z)+\mu_2 w(1+a_2u-w),\quad &x\in\Omega, t>0,\\ z_t=\Delta z-\alpha_2 z+\beta_2u,\quad &x\in \Omega, t>0,\\ \end{array} \right. \end{eqnarray*} in an arbitrary smooth bounded domain $\Omega\subset \mathbb{R}^n$ under homogeneous Neumann boundary conditions. The parameters in the system are positive.We first prove that if $n\leq3$, the corresponding initial-boundary value problem admits a unique global bounded classical solution, under the assumption that $\chi, \xi$, $\mu_i, a_i, \alpha_i$ and $\beta_i(i=1,2)$ satisfy some suitable conditions. Subsequently, we also analyse the asymptotic behavior of solutions to the above system and show that $\bullet$ when $a_1<1$ and both $\frac{\mu_1}{\chi^2}$ and $\frac{\mu_2}{\xi^2}$ are sufficiently large, the global solution $(u, v, w, z)$ of this system exponentially converges to $(\frac{1-a_1}{1+a_1a_2}, \frac{\beta_1(1+a_2)}{\alpha_1(1+a_1a_2)}, \frac{1+a_2}{1+a_1a_2}, \frac{\beta_2(1-a_1)}{\alpha_2(1+a_1a_2)})$ as $t\rightarrow \infty$; $\bullet$ when $a_1>1$ and $\frac{\mu_2}{\xi^2}$ is sufficiently large, the global bounded classical solution $(u, v, w, z)$ of this system exponentially converges to $(0, \frac{\alpha_1}{\beta_1}, 1, 0)$ as $t\rightarrow \infty$; $\bullet$ when $a_1=1$ and $\frac{\mu_2}{\xi^2}$ is sufficiently large, the global bounded classical solution $(u, v, w, z)$ of this system polynomially converges to $(0, \frac{\alpha_1}{\beta_1}, 1, 0)$ as $t\rightarrow \infty$.
 THE GLOBAL EXISTENCE AND A DECAY ESTIMATE OF SOLUTIONS TO THE PHAN-THEIN-TANNER MODEL Ruiying WEI, Yin LI, Zheng-an YAO Acta mathematica scientia,Series B. 2022, 42 (3):  1058-1080.  DOI: 10.1007/s10473-022-0314-6 In this paper, we study the global existence and decay rates of strong solutions to the three dimensional compressible Phan-Thein-Tanner model. By a refined energy method, we prove the global existence under the assumption that the $H^3$ norm of the initial data is small, but that the higher order derivatives can be large. If the initial data belong to homogeneous Sobolev spaces or homogeneous Besov spaces, we obtain the time decay rates of the solution and its higher order spatial derivatives. Moreover, we also obtain the usual $L^p-L^2(1\leq p\leq2)$ type of the decay rate without requiring that the $L^p$ norm of initial data is small.
 THE TIME DECAY RATES OF THE CLASSICAL SOLUTION TO THE POISSON-NERNST-PLANCK-FOURIER EQUATIONS IN $\mathbb{R}^3$ Leilei TONG, Zhong TAN, Xu ZHANG Acta mathematica scientia,Series B. 2022, 42 (3):  1081-1102.  DOI: 10.1007/s10473-022-0315-5 In this work, the Poisson-Nernst-Planck-Fourier system in three dimensions is considered. For when the initial data regards a small perturbation around the constant equilibrium state in a $H^3\cap\dot{H}^{-s} (0\leq s\leq 1/2)$ norm, we obtain the time convergence rate of the global solution by a regularity interpolation trick and an energy method.
 DK SPACES AND CARLESON MEASURES Dongxing LI, Hasi WULAN, Ruhan ZHAO Acta mathematica scientia,Series B. 2022, 42 (3):  1103-1112.  DOI: 10.1007/s10473-022-0316-4 We give some characterizations of Carleson measures for Dirichlet type spaces by using Hadamard products. We also give a one-box condition for such Carleson measures.
 A FRACTIONAL CRITICAL PROBLEM WITH SHIFTING SUBCRITICAL PERTURBATION Qi LI, Chang-Lin XIANG Acta mathematica scientia,Series B. 2022, 42 (3):  1113-1124.  DOI: 10.1007/s10473-022-0317-3 In this paper, we consider a class of fractional problem with subcritical perturbation on a bounded domain as follows: \begin{equation*} (P_{k})\quad \left\{ \begin{array}{ll} \displaystyle (-\Delta)^s u=g(x)[(u-k)^+]^{q-1}+u^{2^{*}_{s}-1},\ \ &x\in \Omega,\\ \displaystyle u>0,\ \ &x\in \Omega,\\ \displaystyle u=0,\ \ &x\in \mathbb{R}^N\backslash \Omega. \end{array} \right. \end{equation*} We prove the existence of nontrivial solutions $u_{k}$ of $(P_{k})$ for each $k\in (0,\infty)$. We also investigate the concentration behavior of the solutions $u_{k}$ as $k\to \infty$.
 THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL Jinlan TAN, Yongyong LI, Chunlei TANG Acta mathematica scientia,Series B. 2022, 42 (3):  1125-1140.  DOI: 10.1007/s10473-022-0318-2 In this paper, we investigate a class of nonlinear Chern-Simons-Schrödinger systems with a steep well potential. By using variational methods, the mountain pass theorem and Nehari manifold methods, we prove the existence of a ground state solution for λ > 0 large enough. Furthermore, we verify the asymptotic behavior of ground state solutions as λ → +∞.
 CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT Mingzhan HUANG, Shouzong LIU, Xinyu SONG, Xiufen ZOU Acta mathematica scientia,Series B. 2022, 42 (3):  1141-1159.  DOI: 10.1007/s10473-022-0319-1 This paper mainly studies the stochastic character of tumor growth in the presence of immune response and periodically pulsed chemotherapy. First, a stochastic impulsive model describing the interaction and competition among normal cells, tumor cells and immune cells under periodically pulsed chemotherapy is established. Then, sufficient conditions for the extinction, non-persistence in the mean, weak and strong persistence in the mean of tumor cells are obtained. Finally, numerical simulations are performed which not only verify the theoretical results derived but also reveal some specific features. The results show that the growth trend of tumor cells is significantly affected by the intensity of noise and the frequency and dose of drug deliveries. In clinical practice, doctors can reduce the randomness of the environment and increase the intensity of drug input to inhibit the proliferation and growth of tumor cells.
 A COMPACTNESS THEOREM FOR STABLE FLAT $SL(2,\mathbb{C})$ CONNECTIONS ON 3-FOLDS Teng HUANG Acta mathematica scientia,Series B. 2022, 42 (3):  1160-1172.  DOI: 10.1007/s10473-022-0320-8 Let $Y$ be a closed $3$-manifold such that all flat $SU(2)$-connections on $Y$ are non-degenerate. In this article, we prove a Uhlenbeck-type compactness theorem on $Y$ for stable flat $SL(2,\mathbb{C})$ connections satisfying an $L^{2}$-bound for the real curvature. Combining the compactness theorem and a result from [7], we prove that the moduli space of the stable flat $SL(2,\mathbb{C})$ connections is disconnected under certain technical assumptions.
 LEARNING RATES OF KERNEL-BASED ROBUST CLASSIFICATION Shuhua WANG, Baohuai SHENG Acta mathematica scientia,Series B. 2022, 42 (3):  1173-1190.  DOI: 10.1007/s10473-022-0321-7 This paper considers a robust kernel regularized classification algorithm with a non-convex loss function which is proposed to alleviate the performance deterioration caused by the outliers. A comparison relationship between the excess misclassification error and the excess generalization error is provided; from this, along with the convex analysis theory, a kind of learning rate is derived. The results show that the performance of the classifier is effected by the outliers, and the extent of impact can be controlled by choosing the homotopy parameters properly.
 BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES Qianjun HE, Xinfeng WU, Dunyan YAN Acta mathematica scientia,Series B. 2022, 42 (3):  1191-1208.  DOI: 10.1007/s10473-022-0322-6 In this paper, we study a boundedness property of the Adams type for multilinear fractional integral operators with the multilinear $L^{r^{\prime},\alpha}$-Hörmander condition and their commutators with vector valued BMO functions on a Morrey space and a predual Morrey space. Moreover, we give an endpoint estimate for multilinear fractional integral operators. As an application, we obtain the boundedness of multilinear Fourier multipliers with limited Sobolev regularity on a Morrey space.
 EXISTENCE RESULTS FOR SINGULAR FRACTIONAL p-KIRCHHOFF PROBLEMS Mingqi XIANG, Vicenţiu D. RǍDULESCU, Binlin ZHANG Acta mathematica scientia,Series B. 2022, 42 (3):  1209-1224.  DOI: 10.1007/s10473-022-0323-5 This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional $p$-Laplacian operator. More precisely, we study the following nonlocal problem: \begin{align*} \begin{cases} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|x|^{\alpha_1p}|y|^{\alpha_2p}|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}{\rm d}x{\rm d}y\right) \mathcal{L}^{s}_pu= |x|^{\beta} f(u)\,\, \ &{\rm in}\ \Omega,\\ u=0\ \ \ \ &{\rm in}\ \mathbb{R}^N\setminus \Omega, \end{cases} \end{align*} where $\mathcal{L}^{s}_p$ is the generalized fractional $p$-Laplacian operator, $N\geq1$, $s\in(0,1)$, $\alpha_1,\alpha_2,\beta\in\mathbb{R}$, $\Omega\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary, and $M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0$, $f:\Omega\rightarrow\mathbb{R}$ are continuous functions. Firstly, we introduce a variational framework for the above problem. Then, the existence of least energy solutions is obtained by using variational methods, provided that the nonlinear term $f$ has $(\theta p-1)$-sublinear growth at infinity. Moreover, the existence of infinitely many solutions is obtained by using Krasnoselskii's genus theory. Finally, we obtain the existence and multiplicity of solutions if $f$ has $(\theta p-1)$-superlinear growth at infinity. The main features of our paper are that the Kirchhoff function may vanish at zero and the nonlinearity may be singular.
 CONTINUOUS SELECTIONS OF THE SET-VALUED METRIC GENERALIZED INVERSE IN 2-STRICTLY CONVEX BANACH SPACES Shaoqiang SHANG Acta mathematica scientia,Series B. 2022, 42 (3):  1225-1237.  DOI: 10.1007/s10473-022-0324-4 In this paper, we prove that if $X$ is an almost convex and 2-strictly convex space, linear operator $T: X \to Y$ is bounded, $N(T)$ is an approximative compact Chebyshev subspace of $X$ and $R(T)$ is a 3-Chebyshev hyperplane, then there exists a homogeneous selection ${T^\sigma }$ of ${T^\partial }$ such that continuous points of ${T^\sigma }$ and ${T^\partial }$ are dense on $Y$.
 A ROBUST COLOR EDGE DETECTION ALGORITHM BASED ON THE QUATERNION HARDY FILTER Wenshan BI, Dong CHENG, Wankai LIU, Kit Ian KOU Acta mathematica scientia,Series B. 2022, 42 (3):  1238-1260.  DOI: 10.1007/s10473-022-0325-3 This paper presents a robust filter called the quaternion Hardy filter (QHF) for color image edge detection. The QHF can be capable of color edge feature enhancement and noise resistance. QHF can be used flexibly by selecting suitable parameters to handle different levels of noise. In particular, the quaternion analytic signal, which is an effective tool in color image processing, can also be produced by quaternion Hardy filtering with specific parameters. Based on the QHF and the improved Di Zenzo gradient operator, a novel color edge detection algorithm is proposed; importantly, it can be efficiently implemented by using the fast discrete quaternion Fourier transform technique. From the experimental results, we conclude that the minimum PSNR improvement rate is 2.3% and the minimum SSIM improvement rate is 30.2% on the CSEE database. The experiments demonstrate that the proposed algorithm outperforms several widely used algorithms.