关键词：
The linearized Moser-Trudinger problem;The first eigenvalue;The first eigenfunction;Asymptotic behavior
摘要：
We revisit the following Moser-Trudinger problem {−Δu=λueu2in Ω,u>0in Ω,u=0on ∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} -\Delta u=\lambda ue^{u^{2}} &\text{in } \Omega , \\ u>0&\text{in } \Omega , \\ u=0 &\text{on } \partial \Omega , \end{cases} $$\end{document} where Ω⊂R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega \subset \mathbb{R}^{2}$\end{document} is a smooth bounded domain and λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda >0$\end{document} is sufficiently small. Qualitative analysis of peaked solutions for Moser-Trudinger type equation in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{R}^{2}$\end{document} has been widely studied in recent decades. In this paper, we continue to consider the qualitative properties of the eigenvalues and eigenfunctions for the corresponding linearized Moser-Trudinger problem by using a variety of local Pohozaev identities combined with some elliptic theory in dimension two. Here we give some fine estimates for the first eigenvalue and eigenfunction of the linearized Moser-Trudinger problem. Since this problem is a critical exponent for dimension two and will lose compactness, we have to obtain some new and technical estimates.
期刊：
Complex Analysis and Operator Theory,2023年17(8):1-18 ISSN：1661-8254
通讯作者：
Chen, ML
作者机构：
[Zheng, Jia] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Zheng, Jia] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.;[Chen, Ming-Liang] Gannan Normal Univ, Sch Math & Comp Sci, Ganzhou 341000, Jiangxi, Peoples R China.
通讯机构：
[Chen, ML ] G;Gannan Normal Univ, Sch Math & Comp Sci, Ganzhou 341000, Jiangxi, Peoples R China.
关键词：
Iterated function system;Self-affine measure;Orthogonality;Spectrality
摘要：
Ever since Jorgensen and Pedersen (J Anal Math 75:185-228, 1998) discovered the first singular spectral measure, the spectral and non-spectral problems of fractal measures have received a lot of attention in recent years. In this work, we study the planar self-affine measure
$$\mu _{M,D}$$
generated by an expanding matrix
$$M\in M_2(\mathbb {Z})$$
and a collinear digit set
$$D=\{0,d_1,d_2,d_3\}\varvec{v}$$
, where
$$\varvec{v}\in \mathbb {Z}^2\backslash \{\varvec{0}\}$$
and
$$d_1,d_2,d_3$$
are different non-zero integers. For the case that
$$\{\varvec{v},M\varvec{v}\}$$
is linearly dependent, the sufficient and necessary condition for
$$\mu _{M,D}$$
to be a spectral measure is given. Moreover, we estimate the number of orthogonal exponential functions in
$$L^2(\mu _{M,D})$$
and give the exact maximal cardinality when
$$\mu _{M,D}$$
is a non-spectral measure. At the same time, partial results are also obtained for the case that
$$\{\varvec{v},M\varvec{v}\}$$
is linearly independent.
作者机构：
[Ye, Jianguo] Kashi Univ, Res Ctr Modern Math & Its Applicat, Sch Math & Stat, Kashi 844000, Peoples R China.;[Yan, Guozheng] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Guozheng Yan] S;School of Mathematics and Statistics, Central China Normal University, Wuhan, China
摘要：
We consider the interior inverse scattering problem for recovering the shape of a penetrable partially coated cavity with external obstacles from the knowledge of measured scattered waves due to point sources. In the first part, we obtain the well-posedness of the direct scattering problem by the variational method. In the second part, we establish the mathematical basis of the linear sampling method to recover both the shape of the cavity, and the shape of the external obstacle, however the exterior transmission eigenvalue problem also plays a key role in the discussion of this paper.
作者机构：
[Yu, Yuantian; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Zhang, Huihui; Zhang, HH] Luoyang Normal Univ, Dept Math, Luoyang 471934, Peoples R China.
通讯机构：
[Zhang, HH ] L;Luoyang Normal Univ, Dept Math, Luoyang 471934, Peoples R China.
摘要：
Given a graph G and a real number alpha is an element of[0, 1], Nikiforov (2017) proposed the A(alpha)-matrix of Gas A(alpha)(G) = alpha D(G) +(1 - alpha) A( G), where A(G) and D( G) are the adjacency matrix and the degree diagonal matrix of G, respectively. The largest eigenvalue of A(alpha)(G), written as lambda(alpha)(G), is called the A alpha-index of G. A set of cycles in a graph G is called independent if no two cycles in it have a common vertex in G. For n > 2k - 1, let S-n,S- 2k-1 be the join of a clique on 2k - 1vertices with an independent set of n - (2k - 1) vertices. The famous Erdos-Posa theorem shows that for k >= 2 and n >= 24k, every n-vertex graph G with at least (2k - 1)(n - k) edges contains kindependent cycles, unless G congruent to S-n,S- 2k-1. In this paper, we consider an A(alpha)-spectral version of this theorem. We show that for fixed k >= 1, 0 < alpha < 1and n >= 104k(3)/alpha(a)(1- alpha), if an n-vertex graph Gsatisfies lambda(alpha)(G) >= lambda(alpha)(S-n,S- 2k-1), then it contains kindependent cycles, unless G congruent to Sn, 2k-1. This extends the result of Zhai and Liu (2022), in which they obtained the adjacency spectral version of the Erd.os-Posa theorem. (c) 2023 Elsevier B.V. All rights reserved.
作者机构：
[Qin, Hong; Xiao, Yao] Zhongnan Univ Econ & Law, Sch Stat & Math, Wuhan, Peoples R China.;[Qin, Hong; Wang, Shiqi; Ning, Jianhui] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.
通讯机构：
[Jianhui Ning] S;School of Mathematics and Statistics, Central China Normal University, Wuhan, People's Republic of China
关键词：
Weighted uniform designs;sequential designs;weighted discrepancy;Gaussian process model
摘要：
Uniform designs seek to distribute design points uniformly in the experimental domain. Some discrepancies have been developed to measure the uniformity by treating all factors equally. It is reason-able when there exists no prior information about the system or when the potential model is completely unclear. However, in the sit-uation of sequential designs, experimental information, such as the importance of each factor, would be obtained from previous stage experiments. With this fact, the weighted L-2-discrepancy is more suit-able than the original discrepancy for choosing follow-up designs. In this paper, the sequentially weighted uniform design is proposed, which is obtained by minimizing the weighted L-2-discrepancy. The weights, indicating the relative importance of each factor, are esti-mated through a Bayesian hierarchical Gaussian process method based on serial experimental data. Results from several classic com-puter simulator examples, as well as a real application in circuit design, demonstrate that the performance of our new method sur-passes that of its counterparts.
期刊：
Advances in Mathematics,2023年434:109320 ISSN：0001-8708
通讯作者：
Lim, MF
作者机构：
[Lei, Antonio] Univ Ottawa, Dept Math & Stat, 150 Louis Pasteur Pvt, Ottawa, ON K1N 6N5, Canada.;[Lim, MF; Lim, Meng Fai] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Lim, MF; Lim, Meng Fai] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Mueller, Katharina] Univ Laval, Dept Math & Stat, Pavill Alexandre Vachon,1045 Ave Med, Quebec City, PQ G1V 0A6, Canada.
通讯机构：
[Lim, MF ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
关键词：
Anticyclotomic Iwasawa theory;Elliptic curves;Supersingular primes;Tate-Shafarevich groups
摘要：
Let p >= 5 be a prime number and E/Q an elliptic curve with good supersingular reduction at p. Under the generalized Heegner hypothesis, we investigate the p-primary subgroups of the Tate-Shafarevich groups of E over number fields contained inside the anticyclotomic Z(p)-extension of an imaginary quadratic field where p splits. (c) 2023 Elsevier Inc. All rights reserved.
期刊：
Advances in Mathematics,2023年431:109257 ISSN：0001-8708
通讯作者：
Lai, CK
作者机构：
[An, Lixiang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[An, Lixiang] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Lai, Chun-Kit] San Francisco State Univ, Dept Math, 1600 Holloway Ave, San Francisco, CA 94132 USA.
通讯机构：
[Lai, CK ] S;San Francisco State Univ, Dept Math, 1600 Holloway Ave, San Francisco, CA 94132 USA.
关键词：
Pro duct-form;Hadamard triples;Self-similar measures;Spectral measures
摘要：
In a previous work by Laba and Wang, it was proved that whenever there is a Hadamard triple (N, D, L), then the associated one-dimensional self-similar measure mu N,D generated by maps N-1(x + d) with d is an element of D, is a spectral measure. In this paper, we introduce pro duct-form digit sets for finitely many Hadamard triples (N, Ak, Lk) by putting each triple into different scales of N. Our main result is to prove that the associated self-similar measure mu N,D is a spectral measure. This result allows us to show that pro duct-form self-similar tiles are spectral sets as long as the tiles in the group ZN obey the Coven-Meyerowitz (T 1), (T 2) tiling condition. Moreover, we show that all self-similar tiles with N = p alpha q are spectral sets, answering a question by Fu, He and Lau in 2015. Finally, our results allow us to offer new singular spectral measures not generated by a single Hadamard triple. Such new examples allow us to classify all spectral self-similar measures generated by four equi-contraction maps, which will appear in a forthcoming paper. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC license (http:// creativecommons .org /licenses /by -nc /4 .0/).
期刊：
Mathematical Methods in the Applied Sciences,2023年46(2):2275-2287 ISSN：0170-4214
通讯作者：
Zhiyuan Li<&wdkj&>Zhiyuan Li Zhiyuan Li Zhiyuan Li
作者机构：
[Jiang, Daijun] Cent China Normal Univ, Hubei Key Lab Math Sci, Sch Math & Stat, Wuhan, Peoples R China.;[Li, Zhiyuan] Ningbo Univ, Sch Math & Stat, 818 Fenghua Rd, Ningbo 315211, Zhejiang, Peoples R China.;[Pauron, Matthieu] ENS Rennes, Bruz, France.;[Yamamoto, Masahiro] Univ Tokyo, Grad Sch Math Sci, Tokyo, Japan.;[Yamamoto, Masahiro] Acad Romanian Scientists, Bucharest, Romania.
通讯机构：
[Zhiyuan Li; Zhiyuan Li Zhiyuan Li Zhiyuan Li] S;School of Mathematics and Statistics, Ningbo University, 818 Fenghua Road, Zhejiang, Ningbo, China
摘要：
In this article, we discuss a solution to time-fractional diffusion equation partial differential t alpha(u-u0)+Au=0$$ {\partial}_t<^>{\alpha}\left(u-{u}_0\right)+ Au=0 $$ with the homogeneous Dirichlet boundary condition, where an elliptic operator -A$$ -A $$ is not necessarily symmetric. We prove that the solution u$$ u $$ is identically zero if its normal derivative with respect to the operator A$$ A $$ vanishes on an arbitrarily chosen subboundary of the spatial domain over a time interval. The proof is based on the Laplace transform and the spectral decomposition for a nonsymmetric elliptic operator. As a direct application, we prove the uniqueness result for an inverse problem on determining the spatial component in the source term by Neumann boundary data on subdoundary.
期刊：
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY,2023年376(05):3063-3085 ISSN：0002-9947
作者机构：
[Yau, Shing-Tung] Harvard Univ, Dept Math, Cambridge, MA 02138 USA.;[Zhao, Quanting] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Zhao, Quanting] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Zheng, Fangyang] Chongqing Normal Univ, Sch Math Sci, Chongqing 401331, Peoples R China.
摘要：
In this paper, we study a special type of compact Hermitian manifolds that are Strominger Ka.hler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Ka.hler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Ka.hler manifold. Previously, we have shown that any SKL manifold (Mn, g) is always pluriclosed, and when the manifold is compact and g is not Ka.hler, it cannot admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n = 2, the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-Ka.hler SKL manifolds in dimension 3 and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, given any compact non-Ka.hler SKL manifold, its Ka.hler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal Ka.hler when n >= 3, and the manifold does not admit any Hermitian symplectic metric.
期刊：
Journal of Algebra,2023年636:42-74 ISSN：0021-8693
通讯作者：
Wang, Q
作者机构：
[Guo, Hongyan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Guo, Hongyan] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Li, Haisheng] Rutgers State Univ, Dept Math Sci, Camden, NJ 08102 USA.;[Wang, Qing; Tan, Shaobin] Xiamen Univ, Sch Math Sci, Xiamen 361005, Peoples R China.
通讯机构：
[Wang, Q ] X;Xiamen Univ, Sch Math Sci, Xiamen 361005, Peoples R China.
摘要：
In this paper, we study a family of infinite-dimensional Lie algebras X ⠂S, where X stands for the type: A, B, C, D, and S is an abelian group, which generalize the A, B, C, D series of trigonometric Lie algebras. Among the main results, we identify X ⠂S with what are called the covariant algebras of the affine Lie algebra L ⠃S with respect to some automorphism groups, where LS is an explicitly defined associative algebra X ⠂S- viewed as a Lie algebra. We then show that restricted modules of level $ naturally correspond to equivariant quasi modules for affine vertex algebras related to LS. Furthermore, for any finite cyclic group S, we completely determine the structures of these four families of Lie algebras, showing that they are essentially affine Kac-Moody Lie algebras of certain types.& COPY; 2023 Elsevier Inc. All rights reserved.
作者机构：
[Qi, Yingfan; Cao, Rongjun; Chen, Minghua; Shi, Jiankang] Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Peoples R China.;[Yin, Xiaobo] Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Chen, MH ] L;Lanzhou Univ, Sch Math & Stat, Lanzhou 730000, Peoples R China.
关键词：
Asymptotically compatible scheme;Nonlocal model;Shifted-symmetric collocation;Stability and convergence analysis
摘要：
The piecewise quadratic polynomial collocation is used to approximate the nonlocal model, which generally leads to a nonsymmetric indefinite system (Chen et al. (2021) [5]). In this case, the discrete maximum principle is not satisfied, which might be trickier for the stability analysis of the high-order numerical schemes (D'Elia et al. (2020) [10]; Leng et al. (2021) [26]). Here, we present a modified (shifted-symmetric) piecewise quadratic polynomial collocation for solving the linear nonlocal diffusion model, which leads to a symmetric positive definite system and satisfies the discrete maximum principle. Using Faulhaber's formula and Riemann zeta function, the perturbation error for symmetric positive definite system and nonsymmetric indefinite system are given. Then rigorous convergence analysis for the nonlocal models are provided under the general horizon parameter delta = O (h beta), with beta >= 0. More concretely, the global error is O (hmin{2,1+beta}) if delta is not set as a grid point, while it recovers O (hmax{2,4-2 beta}) when delta is set as a grid point. We also prove that the shifted-symmetric scheme is asymptotically compatible, which has the global error O (hmin{2,2 beta}) as delta, h -> 0. The numerical experiments (including two-dimensional case) are performed to verify the convergence.(c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.
期刊：
Complex Analysis and Operator Theory,2023年17(1):1-17 ISSN：1661-8254
通讯作者：
Min-Min Zhang
作者机构：
[Zhang, Min-Min; Wei, Saidi] Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Min-Min Zhang] H;Hubei Key Laboratory of Mathematical Sciences, School of Mathematics and Statistics, Central China Normal University, Wuhan, People’s Republic of China
摘要：
Let
$$\mu _{M,D}$$
be the self-affine measure associated with an expanding integer matrix
$$M=\left( \begin{array}{cc} p &{} 0 \\ 0 &{} q \\ \end{array}\right) $$
and
$$D=\left\{ \,\,\begin{pmatrix} 0\\ 0 \end{pmatrix},\,\,\,\begin{pmatrix} 1\\ 1 \end{pmatrix} \,\,\right\} $$
, where |p| and |q| are distinct odd bigger than 1. Such a measure is the simplest and the most important case in the study of the spectral property of self-affine measures with two-elements digit sets, which is an open problem up to now. In this paper, we first construct two classes of 4-element orthogonal exponentials in the corresponding Hilbert space
$$L^2(\mu _{M,D})$$
. Moreover, we prove that, under certain conditions, the constructed 4-element orthogonal exponentials is maximal.
期刊：
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS,2023年43(1):482-506 ISSN：1078-0947
通讯作者：
Angela Pistoia
作者机构：
[Chen, Haixia] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.;[Pistoia, Angela] Sapienza Univ Roma, Dipartimento Sci Base & Applicate Ingn, Rome, Italy.;[Vaira, Giusi] Univ Bari Aldo Moro, Dipartimento Matemat, Bari, Italy.
通讯机构：
[Angela Pistoia] D;Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Italy
摘要：
In this article, we investigate the asymptotic behavior of solutions for a non-autonomous regularized magnetohydrodynamics equations on 3D bounded domains. More precisely, the upper bounds on the number of determining modes and determining nodes for the system are established. The results show that the asymptotic behavior of the weak solution can be determined completely by its first finite number of Fourier modes and the large time behavior of the strong solution can be determined by its values on a finite number of points.
期刊：
BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY,2023年108(2):187-199 ISSN：0004-9727
通讯作者：
MINJIE ZHANG
作者机构：
[Miao, Shujing; LI, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Zhang, Minjie] Hubei Univ Arts & Sci, Sch Math & Stat, Xiangyang 441053, Peoples R China.
通讯机构：
[MINJIE ZHANG] S;School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang 441053, PR China
摘要：
We first establish a lower bound on the size and spectral radius of a graph G to guarantee that G contains a fractional perfect matching. Then, we determine an upper bound on the distance spectral radius of a graph G to ensure that G has a fractional perfect matching. Furthermore, we construct some extremal graphs to show all the bounds are best possible.
作者机构：
[Yuan, Ganghua] Northeast Normal Univ, Sch Math & Stat, KLAS, Changchun 130024, Jilin, Peoples R China.;[Zhao, Yue; Zhao, Y] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Zhao, Y ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
期刊：
Journal of Functional Analysis,2023年284(6):109816 ISSN：0022-1236
通讯作者：
Xiaohua Yao
作者机构：
[Li, Ping] Yangtze Univ, Sch Informat & Math, Jingzhou 434000, Peoples R China.;[Soffer, Avy] Rutgers State Univ, Math Dept, New Brunswick, NJ 08903 USA.;[Yao, Xiaohua] Cent China Normal Univ, Dept Math, Wuhan 430079, Peoples R China.;[Yao, Xiaohua] Cent China Normal Univ, Hubei Prov Key Lab Math Phys, Wuhan 430079, Peoples R China.
通讯机构：
[Xiaohua Yao] D;Department of Mathematics and Hubei Province Key Laboratory of Mathematical Physics, Central China Normal University, Wuhan, 430079, China
关键词：
Decay estimates;Fourth-order Schr?dinger operators;Asymptotic expansion of resolvent;Dimension two
摘要：
In this paper we study the decay estimates of the fourth order Schrodinger operator H = Delta 2 + V(x) on R2 with a bounded decaying potential V(x). We first deduce the asymptotic expansions of resolvent of H near zero threshold in the presence of resonances or eigenvalue, and then use them to establish the L1 - L infinity decay estimates of e-itH generated by the fourth order Schrodinger operator H. Our methods used in the decay estimates depend on Littlewood-Paley decomposition and oscillatory integral theory. Moreover, we also classify these zero resonances as the distributional solutions of H phi = 0 in suitable weighted spaces. Due to the degeneracy of Delta 2 at zero threshold, we remark that the asymptotic expansions of resolvent RV (lambda 4) and the classifications of resonances are much more involved than Schrodinger operator -Delta + V in dimension two.(c) 2022 Elsevier Inc. All rights reserved.
期刊：
Journal of Mathematical Physics,2023年64(1):011506 ISSN：0022-2488
作者机构：
[An, Xiaoming] Guizhou Univ Finance & Econ, Sch Math & Stat, Guiyang 550025, Peoples R China.;[Yang, Xian] Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
摘要：
In this paper, we show a connection between fractional Schrodinger equations with power-law nonlinearity and fractional Schrodinger equations with logarithm-law nonlinearity. We prove that ground state solutions of power-law fractional equations, as p & RARR; 2(+), converge to a ground state solution of logarithm-law fractional equations. In particular, we provide a new proof to the existence of a ground state of logarithm-law fractional Schrodinger equations.
作者机构：
[Liu, Zhongyuan] Henan Univ, Sch Math & Stat, Kaifeng 475004, Peoples R China.;[Luo, Peng] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Luo, Peng] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构：
[Peng Luo] S;School of Mathematics and Statistics and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, China
摘要：
In this paper, we study the following critical elliptic problem with a variable exponent:
$$\left\{ {\matrix{{ - \Delta u = {u^{p + \epsilon a\left( x \right)}}} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {u > 0} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm{on}}\,\partial \Omega ,} \hfill \cr } } \right.$$
where
$$a\left( x \right) \in {C^2}\left( {\overline \Omega } \right),\,p = {{N + 2} \over {N - 2}},\,\,\epsilon > 0$$
, and Ω is a smooth bounded domain in ℝN (N ≽ 4). We show that for ∊ small enough, there exists a family of bubble solutions concentrating at the negative stable critical point of the function a(x). This is a new perturbation to the critical elliptic equation in contrast to the usual subcritical or supercritical perturbation, and gives the first existence result for the critical elliptic problem with a variable exponent.
