Forum Mathematicum最新文献

{"title":"Normalized solutions for the fractional Schrödinger equation with combined nonlinearities","authors":"Shengbing Deng, Qiaoran Wu","doi":"10.1515/forum-2023-0424","DOIUrl":"https://doi.org/10.1515/forum-2023-0424","url":null,"abstract":"In this paper, we study the normalized solutions for the following fractional Schrödinger equation with combined nonlinearities <jats:disp-formula-group> <jats:disp-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing=\"0pt\" rowspacing=\"0pt\"> <m:mtr> <m:mtd columnalign=\"right\"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi>s</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mtd> <m:mtd columnalign=\"left\"> <m:mrow> <m:mi /> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>μ</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo fence=\"true\" stretchy=\"false\">|</m:mo> <m:mi>u</m:mi> <m:mo fence=\"true\" stretchy=\"false\">|</m:mo> </m:mrow> <m:mrow> <m:mi>q</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo fence=\"true\" stretchy=\"false\">|</m:mo> <m:mi>u</m:mi> <m:mo fence=\"true\" stretchy=\"false\">|</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd /> <m:mtd columnalign=\"right\"> <m:mrow> <m:mrow> <m:mtext>in </m:mtext> <m:mo>⁢</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"right\"> <m:mrow> <m:mstyle displaystyle=\"true\"> <m:msub> <m:mo largeop=\"true\" symmetric=\"true\">∫</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:msub> </m:mstyle> <m:mrow> <m:mpadded width=\"+1.7pt\"> <m:msup> <m:mi>u</m:mi> <m:mn>2</m:mn> </m:msup> </m:mpadded> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign=\"left\"> <m:mrow> <m:mrow> <m:mi /> <m:mo>=</m:mo> <m:msup> <m:mi>a</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0424_eq_0161.png\" /> <jats:tex-math>displaystyleleft{begin{aligned} displaystyle{}(-Delta)^{s}u&% displaystyle=lambda u+mulvert urvert^{q-2}u+lvert urvert^{p-2}u&&% displaystylephantom{}text{in }mathbb{R}^{N}, displaystyleint_{mathbb{R}^{N}}u^{2},dx&displaystyle=a^{2},end{aligned}right.</jats:tex-math> </jats:alternatives> </jats:disp-formula> </jats:disp-formula-group> where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>s</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0424_eq_0263.png\" /> <jats:tex-math>{0<s<1}</jats:tex-math> </jats:alternativ","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"190 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656308","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}

{"title":"Profinite genus of fundamental groups of compact flat manifolds with the cyclic holonomy group of square-free order","authors":"Genildo de Jesus Nery","doi":"10.1515/forum-2021-0298","DOIUrl":"https://doi.org/10.1515/forum-2021-0298","url":null,"abstract":"In this article, we study the extent to which an <jats:italic>n</jats:italic>-dimensional compact flat manifold with the cyclic holonomy group of square-free order may be distinguished by the finite quotients of its fundamental group. In particular, we display a formula for the cardinality of profinite genus of the fundamental group of an <jats:italic>n</jats:italic>-dimensional compact flat manifold with the cyclic holonomy group of square-free order.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656232","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}

{"title":"The C*-algebra of the Boidol group","authors":"Ying-Fen Lin, Jean Ludwig","doi":"10.1515/forum-2021-0209","DOIUrl":"https://doi.org/10.1515/forum-2021-0209","url":null,"abstract":"The Boidol group is the smallest non-<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mo>∗</m:mo> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2021-0209_eq_0575.png\" /> <jats:tex-math>{ast}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-regular exponential Lie group. It is of dimension 4 and its Lie algebra is an extension of the Heisenberg Lie algebra by the reals with the roots 1 and -1. We describe the C*-algebra of the Boidol group as an algebra of operator fields defined over the spectrum of the group. It is the only connected solvable Lie group of dimension less than or equal to 4 whose group C*-algebra had not yet been determined.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"291 2 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656067","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}

{"title":"Degrees of generalized Kloosterman sums","authors":"Liping Yang","doi":"10.1515/forum-2023-0295","DOIUrl":"https://doi.org/10.1515/forum-2023-0295","url":null,"abstract":"The modern study of the exponential sums is mainly about their analytic estimates as complex numbers, which is local. In this paper, we study one global property of the exponential sums by viewing them as algebraic integers. For a kind of generalized Kloosterman sums, we present their degrees as algebraic integers.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"258 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645379","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}

{"title":"Algebraic results on rngs of singular functions","authors":"Arran Fernandez, Müge Saadetoğlu","doi":"10.1515/forum-2023-0445","DOIUrl":"https://doi.org/10.1515/forum-2023-0445","url":null,"abstract":"We consider a Mikusiński-type convolution algebra <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>C</m:mi> <m:mi>α</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0445_eq_0144.png\" /> <jats:tex-math>{C_{alpha}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, including functions with power-type singularities at the origin as well as all functions continuous on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0445_eq_0198.png\" /> <jats:tex-math>{[0,infty)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Algebraic properties of this space are derived, including its ideal structure, filtered and graded structure, and Jacobson radical. Applications to operators of fractional calculus and the associated integro-differential equations are discussed.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"37 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645371","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}

{"title":"Bi-parameter and bilinear Calderón–Vaillancourt theorem with critical order","authors":"Jiao Chen, Liang Huang, Guozhen Lu","doi":"10.1515/forum-2023-0458","DOIUrl":"https://doi.org/10.1515/forum-2023-0458","url":null,"abstract":"In this paper, we establish the sharp Calderón–Vaillancourt theorem on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0458_eq_0174.png\" /> <jats:tex-math>L^{p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> spaces for bi-parameter and bilinear pseudo-differential operators with symbols of critical order by deriving a sufficient and necessary condition on its symbol. This sharpens the result of [G. Lu and L. Zhang, Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order, Forum Math. 28 2016, 6, 1087–1094] which was only proved for symbols of subcritical order.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"177 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139645367","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}

{"title":"Hardy inequalities on metric measure spaces, IV: The case p=1","authors":"Michael Ruzhansky, Anjali Shriwastawa, Bankteshwar Tiwari","doi":"10.1515/forum-2023-0319","DOIUrl":"https://doi.org/10.1515/forum-2023-0319","url":null,"abstract":"In this paper, we investigate the two-weight Hardy inequalities on metric measure space possessing polar decompositions for the case &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0319_eq_0172.png\" /&gt; &lt;jats:tex-math&gt;{p=1}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;≤&lt;/m:mo&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;∞&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0319_eq_0087.png\" /&gt; &lt;jats:tex-math&gt;{1leq q&lt;infty}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. This result complements the Hardy inequalities obtained in [M. Ruzhansky and D. Verma, Hardy inequalities on metric measure spaces, Proc. Roy. Soc. A. 475 2019, 2223, Article ID 20180310] in the case &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mo&gt;≤&lt;/m:mo&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;∞&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0319_eq_0086.png\" /&gt; &lt;jats:tex-math&gt;{1&lt;pleq q&lt;infty}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. The case &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0319_eq_0172.png\" /&gt; &lt;jats:tex-math&gt;{p=1}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; requires a different argument and does not follow as the limit of known inequalities for &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mo&gt;&gt;&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0319_eq_0173.png\" /&gt; &lt;jats:tex-math&gt;{p&gt;1}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. As a byproduct, we also obtain the best constant in the established inequality. We give examples obtaining new weighted Hardy inequalities on homogeneous Lie groups, on hyperbolic spaces and on Cartan–Hadamard manifolds for the case &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0319_eq_0172.png\" /&gt; &lt;jats:tex-math&gt;{p=1}","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139471173","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}

{"title":"Transcendence on algebraic groups","authors":"Duc Hiep Pham","doi":"10.1515/forum-2023-0078","DOIUrl":"https://doi.org/10.1515/forum-2023-0078","url":null,"abstract":"In this paper, we give some new results on transcendence on algebraic groups. These results extend some previous ones established on commutative or linear algebraic groups to arbitrary algebraic groups in complex and <jats:italic>p</jats:italic>-adic fields, respectively.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"25 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139421105","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}

{"title":"An explicit version of Bombieri’s log-free density estimate and Sárközy’s theorem for shifted primes","authors":"Jesse Thorner, Asif Zaman","doi":"10.1515/forum-2023-0091","DOIUrl":"https://doi.org/10.1515/forum-2023-0091","url":null,"abstract":"We make explicit Bombieri’s refinement of Gallagher’s log-free “large sieve density estimate near <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>σ</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0091_eq_0510.png\" /> <jats:tex-math>{sigma=1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>” for Dirichlet <jats:italic>L</jats:italic>-functions. We use this estimate and recent work of Green to prove that if <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0091_eq_0355.png\" /> <jats:tex-math>{Ngeq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is an integer, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>A</m:mi> <m:mo>⊆</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0091_eq_0326.png\" /> <jats:tex-math>{Asubseteq{1,ldots,N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and for all primes <jats:italic>p</jats:italic> no two elements in <jats:italic>A</jats:italic> differ by <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0091_eq_0579.png\" /> <jats:tex-math>{p-1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, then <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>A</m:mi> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mo>≪</m:mo> <m:msup> <m:mi>N</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:msup> <m:mn>10</m:mn> <m:mrow> <m:mo>-</m:mo> <m:mn>18</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0091_eq_0630.png\" /> <jats:tex-math>{|A|ll N^{1-10^{-18}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. This strengthens a theorem of Sárközy.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"52 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139422645","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}

{"title":"Multiple normalized solutions for fractional elliptic problems","authors":"Thin Van Nguyen, Vicenţiu D. Rădulescu","doi":"10.1515/forum-2023-0366","DOIUrl":"https://doi.org/10.1515/forum-2023-0366","url":null,"abstract":"In this article, we are first concerned with the existence of multiple normalized solutions to the following fractional &lt;jats:italic&gt;p&lt;/jats:italic&gt;-Laplace problem: &lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mo&gt;{&lt;/m:mo&gt; &lt;m:mtable columnspacing=\"0pt\" displaystyle=\"true\" rowspacing=\"0pt\"&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"right\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;-&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;/m:msubsup&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"script\"&gt;𝒱&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo fence=\"true\" stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo fence=\"true\" stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mo&gt;-&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mtd&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi /&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;λ&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo fence=\"true\" stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo fence=\"true\" stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;m:mo&gt;-&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo separator=\"true\"&gt; &lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mtext&gt;in &lt;/m:mtext&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mtd&gt; &lt;/m:mtr&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"right\"&gt; &lt;m:mrow&gt; &lt;m:msub&gt; &lt;m:mo largeop=\"true\" symmetric=\"true\"&gt;∫&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:msub&gt; &lt;m:mrow&gt; &lt;m:mpadded width=\"+1.7pt\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo fence=\"true\" stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo fence=\"true\" stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:mpadded&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;𝑑&lt;/m:mo&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mtd&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi /&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mi&gt;a&lt;/m:mi&gt; &lt;m:mi&gt;p&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mtd&gt; &lt;/m:mtr&gt; &lt;/m:mtable&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0366_eq_0162.png\" /&gt; &lt;jats:tex-math&gt;left{begin{aligned} displaystyle{}(-Delta)_{p}^{s}v+mathcal{V}(xi x)% lvert vrvert^{p-2}v&amp;displaystyle=lambdalvert vrvert^{p-2}v+f(v)quad% text{in }mathbb{R}^{N}, displaystyleint_{mathbb{R}^{N}}lvert vrvert^{p},dx&amp;displaystyle=a^{p},% end{aligned}right.&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"82 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139423480","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}