Pub Date : 2023-02-01DOI: 10.1017/S001309152300010X
Liqian Jia, Xinfu Li, Shiwang Ma
Abstract We consider the following Kirchhoff-type problem in an unbounded exterior domain $Omegasubsetmathbb{R}^{3}$: (*) begin{align} left{ begin{array}{ll} -left(a+bdisplaystyle{int}_{Omega}|nabla u|^{2},{rm d}xright)triangle u+lambda u=f(u), & xinOmega, u=0, & xinpartial Omega, end{array}right. end{align}where a > 0, $bgeq0$, and λ > 0 are constants, $partialOmeganeqemptyset$, $mathbb{R}^{3}backslashOmega$ is bounded, $uin H_{0}^{1}(Omega)$, and $fin C^1(mathbb{R},mathbb{R})$ is subcritical and superlinear near infinity. Under some mild conditions, we prove that if begin{equation*}-Delta u+lambda u=f(u), qquad xin mathbb R^3 end{equation*}has only finite number of positive solutions in $H^1(mathbb R^3)$ and the diameter of the hole $mathbb R^3setminus Omega$ is small enough, then the problem (*) admits a positive solution. Same conclusion holds true if Ω is fixed and λ > 0 is small. To our best knowledge, there is no similar result published in the literature concerning the existence of positive solutions to the above Kirchhoff equation in exterior domains.
摘要我们考虑无界外域$Omegasubetmathbb｛R｝^｛3｝$中的以下Kirchhoff型问题：{ll}-left（a+bdisplaystyle｛int｝_｛Omega｝|nabla u|^｛2｝，｛rm d｝xright）三角形u+lambda u=f（u），&xinOmega，u=0，&x in partialOmega。end｛array｝right。完{align}wherea>0、$bgeq0$和λ>0是常数，$partialOmeganeqemptyset$、$mathbb｛R｝^｛3｝反斜杠Omega$是有界的，H_｛0｝^｝1｝（Omega）$中的$u和C^1（mathbb{R｝，mathbb R｝）$的$f在无穷大附近是亚临界和超线性的。在一些温和的条件下，我们证明了如果begin｛equation*｝-Delta u+lambda u=f（u），qquad xinmathbb R^3end｛equion*｝在$H^1（mathbb R ^3）$中只有有限个正解，并且孔的直径$mathbb R^3setminusOmega$足够小，那么问题（*）允许正解。如果Ω是固定的并且λ>0很小，则同样的结论成立。据我们所知，关于上述Kirchhoff方程在外域中正解的存在性，文献中没有发表类似的结果。
{"title":"Existence of positive solutions for Kirchhoff-type problem in exterior domains","authors":"Liqian Jia, Xinfu Li, Shiwang Ma","doi":"10.1017/S001309152300010X","DOIUrl":"https://doi.org/10.1017/S001309152300010X","url":null,"abstract":"Abstract We consider the following Kirchhoff-type problem in an unbounded exterior domain $Omegasubsetmathbb{R}^{3}$: (*)\u0000begin{align}\u0000left{\u0000begin{array}{ll}\u0000-left(a+bdisplaystyle{int}_{Omega}|nabla u|^{2},{rm d}xright)triangle u+lambda u=f(u), & xinOmega,\u0000\u0000u=0, & xinpartial Omega,\u0000end{array}right.\u0000end{align}where a > 0, $bgeq0$, and λ > 0 are constants, $partialOmeganeqemptyset$, $mathbb{R}^{3}backslashOmega$ is bounded, $uin H_{0}^{1}(Omega)$, and $fin C^1(mathbb{R},mathbb{R})$ is subcritical and superlinear near infinity. Under some mild conditions, we prove that if begin{equation*}-Delta u+lambda u=f(u), qquad xin mathbb R^3 end{equation*}has only finite number of positive solutions in $H^1(mathbb R^3)$ and the diameter of the hole $mathbb R^3setminus Omega$ is small enough, then the problem (*) admits a positive solution. Same conclusion holds true if Ω is fixed and λ > 0 is small. To our best knowledge, there is no similar result published in the literature concerning the existence of positive solutions to the above Kirchhoff equation in exterior domains.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"66 1","pages":"182 - 217"},"PeriodicalIF":0.7,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44214441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-01DOI: 10.1017/S0013091523000068
Pinka Dey
Abstract Dold manifolds $P(m,n)$ are certain twisted complex projective space bundles over real projective spaces and serve as generators for the unoriented cobordism algebra of smooth manifolds. The paper investigates the structure of finite groups that act freely on products of Dold manifolds. It is proved that if a finite group G acts freely and $ mathbb{Z}_2 $ cohomologically trivially on a finite CW-complex homotopy equivalent to ${prod_{i=1}^{k} P(2m_i,n_i)}$, then $Gcong (mathbb{Z}_2)^l$ for some $lleq k$ (see Theorem A for the exact bound). We also determine some bounds in the case when for each i, ni is even and mi is arbitrary. As a consequence, the free rank of symmetry of these manifolds is determined for cohomologically trivial actions.
{"title":"Free rank of symmetry of products of Dold manifolds","authors":"Pinka Dey","doi":"10.1017/S0013091523000068","DOIUrl":"https://doi.org/10.1017/S0013091523000068","url":null,"abstract":"Abstract Dold manifolds $P(m,n)$ are certain twisted complex projective space bundles over real projective spaces and serve as generators for the unoriented cobordism algebra of smooth manifolds. The paper investigates the structure of finite groups that act freely on products of Dold manifolds. It is proved that if a finite group G acts freely and $ mathbb{Z}_2 $ cohomologically trivially on a finite CW-complex homotopy equivalent to ${prod_{i=1}^{k} P(2m_i,n_i)}$, then $Gcong (mathbb{Z}_2)^l$ for some $lleq k$ (see Theorem A for the exact bound). We also determine some bounds in the case when for each i, ni is even and mi is arbitrary. As a consequence, the free rank of symmetry of these manifolds is determined for cohomologically trivial actions.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"66 1","pages":"117 - 132"},"PeriodicalIF":0.7,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46883159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-01DOI: 10.1017/S0013091523000081
A. Baraviera, P. Duarte, M. J. Torres
Abstract In this paper, we address the issue of synchronization of coupled systems, introducing concepts of local and global synchronization for a class of systems that extend the model of coupled map lattices. A criterion for local synchronization is given; numerical experiments are exhibited to illustrate the criteria and also to raise some questions in the end of the text.
{"title":"Synchronization of coupled map lattices","authors":"A. Baraviera, P. Duarte, M. J. Torres","doi":"10.1017/S0013091523000081","DOIUrl":"https://doi.org/10.1017/S0013091523000081","url":null,"abstract":"Abstract In this paper, we address the issue of synchronization of coupled systems, introducing concepts of local and global synchronization for a class of systems that extend the model of coupled map lattices. A criterion for local synchronization is given; numerical experiments are exhibited to illustrate the criteria and also to raise some questions in the end of the text.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"66 1","pages":"143 - 163"},"PeriodicalIF":0.7,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56897207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.1017/S0013091522000529
Anuj Jakhar, Sumandeep Kaur, Surender Kumar
Abstract Let $K={mathbf {Q}}(theta )$ be an algebraic number field with $theta$ a root of an irreducible polynomial $x^5+ax+bin {mathbf {Z}}[x]$. In this paper, for every rational prime $p$, we provide necessary and sufficient conditions on $a,,~b$ so that $p$ is a common index divisor of $K$. In particular, we give sufficient conditions on $a,,~b$ for which $K$ is non-monogenic. We illustrate our results through examples.
{"title":"Common index divisor of the number fields defined by $x^5+,ax,+b$","authors":"Anuj Jakhar, Sumandeep Kaur, Surender Kumar","doi":"10.1017/S0013091522000529","DOIUrl":"https://doi.org/10.1017/S0013091522000529","url":null,"abstract":"Abstract Let $K={mathbf {Q}}(theta )$ be an algebraic number field with $theta$ a root of an irreducible polynomial $x^5+ax+bin {mathbf {Z}}[x]$. In this paper, for every rational prime $p$, we provide necessary and sufficient conditions on $a,,~b$ so that $p$ is a common index divisor of $K$. In particular, we give sufficient conditions on $a,,~b$ for which $K$ is non-monogenic. We illustrate our results through examples.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"1147 - 1161"},"PeriodicalIF":0.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43563895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.1017/s0013091523000020
{"title":"PEM series 2 volume 65 issue 4 Cover and Back matter","authors":"","doi":"10.1017/s0013091523000020","DOIUrl":"https://doi.org/10.1017/s0013091523000020","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":" ","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44934371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.1017/S0013091522000499
Michael C. Burkhart
Abstract We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime $p$, a Sylow $p$-subgroup of one complement is conjugate to a Sylow $p$-subgroup of the other. As a corollary, we find that any two supersoluble complements of an abelian subgroup $N$ in a finite split extension $G$ are conjugate if and only if, for each prime $p$, there exists a Sylow $p$-subgroup $S$ of $G$ such that any two complements of $Scap N$ in $S$ are conjugate in $G$. In particular, restricting to supersoluble groups allows us to ease D. G. Higman's stipulation that the complements of $Scap N$ in $S$ be conjugate within $S$. We then consider group actions and obtain a fixed point result for non-coprime actions analogous to Glauberman's lemma.
摘要证明了有限分裂扩展上阿贝尔基的两个超溶补共轭当且仅当，对于每一个素数$p$，一个补的Sylow $p$-子群共轭于另一个素数$p$-子群。作为推论，我们发现有限分裂扩展$G$中任意两个阿贝子群$N$的超溶补是共轭的，当且仅当，对于每一个素数$p$，存在$G$的Sylow $p$-子群$S$，使得$S$中$Scap N$的任意两个补在$G$中共轭。特别地，对超溶基团的限制使我们可以简化D. G. Higman关于$S$中$Scap N$的补在$S$内共轭的规定。然后，我们考虑群体行动，并得到了类似于格劳伯曼引理的非互素行动的不动点结果。
{"title":"Conjugacy conditions for supersoluble complements of an abelian base and a fixed point result for non-coprime actions","authors":"Michael C. Burkhart","doi":"10.1017/S0013091522000499","DOIUrl":"https://doi.org/10.1017/S0013091522000499","url":null,"abstract":"Abstract We demonstrate that two supersoluble complements of an abelian base in a finite split extension are conjugate if and only if, for each prime $p$, a Sylow $p$-subgroup of one complement is conjugate to a Sylow $p$-subgroup of the other. As a corollary, we find that any two supersoluble complements of an abelian subgroup $N$ in a finite split extension $G$ are conjugate if and only if, for each prime $p$, there exists a Sylow $p$-subgroup $S$ of $G$ such that any two complements of $Scap N$ in $S$ are conjugate in $G$. In particular, restricting to supersoluble groups allows us to ease D. G. Higman's stipulation that the complements of $Scap N$ in $S$ be conjugate within $S$. We then consider group actions and obtain a fixed point result for non-coprime actions analogous to Glauberman's lemma.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"1075 - 1079"},"PeriodicalIF":0.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46109688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.1017/S0013091522000487
E. Marušić‐Paloka
Abstract We study the asymptotic behaviour of the periodically mixed Zaremba problem. We cover the part of the boundary by a chess board with a small period (square size) $varepsilon$ and impose the Dirichlet condition on black and the Neumann condition on white squares. As $varepsilon to 0$, we get the effective boundary condition which is always of the Dirichlet type. The Dirichlet data on the boundary, however, depend on the ratio between the magnitudes of the two boundary values.
{"title":"Dirichlet vs Neumann","authors":"E. Marušić‐Paloka","doi":"10.1017/S0013091522000487","DOIUrl":"https://doi.org/10.1017/S0013091522000487","url":null,"abstract":"Abstract We study the asymptotic behaviour of the periodically mixed Zaremba problem. We cover the part of the boundary by a chess board with a small period (square size) $varepsilon$ and impose the Dirichlet condition on black and the Neumann condition on white squares. As $varepsilon to 0$, we get the effective boundary condition which is always of the Dirichlet type. The Dirichlet data on the boundary, however, depend on the ratio between the magnitudes of the two boundary values.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"1063 - 1074"},"PeriodicalIF":0.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42846585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.1017/S0013091522000542
Gang Yang, Junpeng Wang
Abstract Let $R$ be a left coherent ring. It is proven that if an $R$-module $M$ has a finite FP-injective dimension, then the Gorenstein projective (resp. Gorenstein flat) dimension and the projective (resp. flat) dimension coincide. Also, we obtain that the pair ($mathcal {GP},, mathcal {GP}^{perp }$) forms a projective cotorsion pair under some mild conditions.
{"title":"FP-injective dimensions and Gorenstein homology","authors":"Gang Yang, Junpeng Wang","doi":"10.1017/S0013091522000542","DOIUrl":"https://doi.org/10.1017/S0013091522000542","url":null,"abstract":"Abstract Let $R$ be a left coherent ring. It is proven that if an $R$-module $M$ has a finite FP-injective dimension, then the Gorenstein projective (resp. Gorenstein flat) dimension and the projective (resp. flat) dimension coincide. Also, we obtain that the pair ($mathcal {GP},, mathcal {GP}^{perp }$) forms a projective cotorsion pair under some mild conditions.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"1183 - 1199"},"PeriodicalIF":0.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49102083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-01DOI: 10.1017/s0013091523000019
{"title":"PEM series 2 volume 65 issue 4 Cover and Front matter","authors":"","doi":"10.1017/s0013091523000019","DOIUrl":"https://doi.org/10.1017/s0013091523000019","url":null,"abstract":"","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":" ","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48560278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-18DOI: 10.1017/S0013091522000451
Yujian Su, Senli Liu
Abstract In this paper, we study the Lieb's translation lemma in Coulomb–Sobolev space and then apply it to investigate the existence of Pohožaev type ground state solution for elliptic equation with van der Waals type potential.
{"title":"Elliptic equation with van der Waals type potential","authors":"Yujian Su, Senli Liu","doi":"10.1017/S0013091522000451","DOIUrl":"https://doi.org/10.1017/S0013091522000451","url":null,"abstract":"Abstract In this paper, we study the Lieb's translation lemma in Coulomb–Sobolev space and then apply it to investigate the existence of Pohožaev type ground state solution for elliptic equation with van der Waals type potential.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":"65 1","pages":"1048 - 1062"},"PeriodicalIF":0.7,"publicationDate":"2022-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44568211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：