Forum Mathematicum最新文献

{"title":"Existence and multiplicity of solutions for fractional Schrödinger-p-Kirchhoff equations in ℝ N","authors":"Huo Tao, Lin Li, Patrick Winkert","doi":"10.1515/forum-2023-0385","DOIUrl":"https://doi.org/10.1515/forum-2023-0385","url":null,"abstract":"This paper concerns the existence and multiplicity of solutions for a nonlinear Schrödinger–Kirchhoff-type equation involving the fractional <jats:italic>p</jats:italic>-Laplace operator in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0385_eq_0416.png\" /> <jats:tex-math>{mathbb{R}^{N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Precisely, we study the Kirchhoff-type problem <jats:disp-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mo maxsize=\"260%\" minsize=\"260%\">(</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:msub> <m:mo largeop=\"true\" symmetric=\"true\">∬</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:msub> <m:mrow> <m:mpadded width=\"+1.7pt\"> <m:mfrac> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mrow> <m:mi>N</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>⁢</m:mo> <m:mi>p</m:mi> </m:mrow> </m:mrow> </m:msup> </m:mfrac> </m:mpadded> <m:mo>⁢</m:mo> <m:mrow> <m:mo>d</m:mo> <m:mpadded width=\"+1.7pt\"> <m:mi>x</m:mi> </m:mpadded> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>d</m:mo> <m:mi>y</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo maxsize=\"260%\" minsize=\"260%\">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msubsup> <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>p</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>V</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>u</m:mi> <m:mo 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:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo mathvariant=\"italic\" separator=\"true\">","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"273 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299679","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 existence of optimal solutions for nonlocal partial systems involving fractional Laplace operator with arbitrary growth","authors":"Siyao Peng","doi":"10.1515/forum-2023-0265","DOIUrl":"https://doi.org/10.1515/forum-2023-0265","url":null,"abstract":"In this paper, we investigate nonlocal partial systems that incorporate the fractional Laplace operator. Our primary focus is to establish a theorem concerning the existence of optimal solutions for these equations. To achieve this, we utilize two fundamental tools: information obtained from an iterative reconstruction algorithm and a variant of the Phragmén–Lindelöf principle of concentration and compactness tailored for fractional systems. By employing these tools, we provide valuable insights into the nature of nonlocal partial systems and their optimal solutions.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300114","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":"On arithmetic quotients of the group SL2 over a quaternion division k-algebra","authors":"Sophie Koch, Joachim Schwermer","doi":"10.1515/forum-2023-0422","DOIUrl":"https://doi.org/10.1515/forum-2023-0422","url":null,"abstract":"Given a totally real algebraic number field <jats:italic>k</jats:italic> of degree <jats:italic>s</jats:italic>, we consider locally symmetric spaces <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>/</m:mo> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0422_eq_0351.png\" /> <jats:tex-math>{X_{G}/Gamma}</jats:tex-math> </jats:alternatives> </jats:inline-formula> associated with arithmetic subgroups Γ of the special linear algebraic <jats:italic>k</jats:italic>-group <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>SL</m:mi> <m:mrow> <m:msub> <m:mi>M</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>D</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0422_eq_0183.png\" /> <jats:tex-math>{G=mathrm{SL}_{M_{2}(D)}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, attached to a quaternion division <jats:italic>k</jats:italic>-algebra <jats:italic>D</jats:italic>. The group <jats:italic>G</jats:italic> is <jats:italic>k</jats:italic>-simple, of <jats:italic>k</jats:italic>-rank one, and non-split over <jats:italic>k</jats:italic>. Using reduction theory, one can construct an open subset <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>Y</m:mi> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:msub> <m:mo>⊂</m:mo> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mi>G</m:mi> </m:msub> <m:mo>/</m:mo> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0422_eq_0361.png\" /> <jats:tex-math>{Y_{Gamma}subset X_{G}/Gamma}</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that its closure <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mover accent=\"true\"> <m:mi>Y</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0422_eq_0583.png\" /> <jats:tex-math>{overline{Y}_{Gamma}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a compact manifold with boundary <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:msub> <m:mover accent=\"true\"> <m:mi>Y</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mi mathvariant=\"normal\">Γ</m:mi> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xli","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"32 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299886","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":"Paley inequality for the Weyl transform and its applications","authors":"Ritika Singhal, N. Shravan Kumar","doi":"10.1515/forum-2023-0302","DOIUrl":"https://doi.org/10.1515/forum-2023-0302","url":null,"abstract":"In this paper, we prove several versions of the classical Paley inequality for the Weyl transform. As for some applications, we prove a version of the Hörmander’s multiplier theorem to discuss <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-2023-0302_eq_0237.png\" /> <jats:tex-math>{L^{p}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-<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>q</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0302_eq_0241.png\" /> <jats:tex-math>{L^{q}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> boundedness of the Weyl multipliers and prove the Hardy–Littlewood inequality. We also consider the vector-valued version of the inequalities of Paley, Hausdorff–Young, and Hardy–Littlewood and their relations. Finally, we also prove Pitt’s inequality for the Weyl transform.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299847","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":"Free groups generated by two unipotent maps","authors":"Chao Jiang, Baohua Xie","doi":"10.1515/forum-2023-0442","DOIUrl":"https://doi.org/10.1515/forum-2023-0442","url":null,"abstract":"Let <jats:italic>A</jats:italic> and <jats:italic>B</jats:italic> be two unipotent elements of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>SU</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <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-2023-0442_eq_0338.png\" /> <jats:tex-math>{mathrm{SU}(2,1)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with distinct fixed points. In [S. B. Kalane and J. R. Parker, Free groups generated by two parabolic maps, Math. Z. 303 2023, 1, Paper No. 9], the authors gave several conditions that guarantee the subgroup <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:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</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-2023-0442_eq_0313.png\" /> <jats:tex-math>{langle A,Brangle}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is discrete and free by using Klein’s combination theorem. We will improve their conditions by using a variant of Klein’s combination theorem. With the same arguments and the additional assumption that <jats:italic>AB</jats:italic> is unipotent, we also extend Parker and Will’s condition that guarantees the subgroup <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:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</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-2023-0442_eq_0313.png\" /> <jats:tex-math>{langle A,Brangle}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is discrete and free in [J. R. Parker and P. Will, A complex hyperbolic Riley slice, Geom. Topol. 21 2017, 6, 3391–3451].","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"35 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300127","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":"Existence of strong solutions for one-dimensional reflected mixed stochastic delay differential equations","authors":"Monir Chadad, Mohamed Erraoui","doi":"10.1515/forum-2023-0288","DOIUrl":"https://doi.org/10.1515/forum-2023-0288","url":null,"abstract":"Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"155 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299674","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":"On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting","authors":"Maria Rosaria Pati","doi":"10.1515/forum-2023-0141","DOIUrl":"https://doi.org/10.1515/forum-2023-0141","url":null,"abstract":"We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form <jats:italic>f</jats:italic> and an imaginary quadratic field satisfying a “relaxed” Heegner hypothesis. Let Λ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo–Vigni, we construct the Λ-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa–Greenberg main conjecture for the <jats:italic>p</jats:italic>-adic <jats:italic>L</jats:italic>-function defined by Magrone.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"21 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300117","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":"Explicit bounds for the solutions of superelliptic equations over number fields","authors":"Attila Bérczes, Yann Bugeaud, Kálmán Győry, Jorge Mello, Alina Ostafe, Min Sha","doi":"10.1515/forum-2023-0381","DOIUrl":"https://doi.org/10.1515/forum-2023-0381","url":null,"abstract":"Let &lt;jats:italic&gt;f&lt;/jats:italic&gt; be a polynomial with coefficients in the ring &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi&gt;O&lt;/m:mi&gt; &lt;m:mi&gt;S&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0381_eq_0397.png\" /&gt; &lt;jats:tex-math&gt;{O_{S}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; of &lt;jats:italic&gt;S&lt;/jats:italic&gt;-integers of a number field &lt;jats:italic&gt;K&lt;/jats:italic&gt;, &lt;jats:italic&gt;b&lt;/jats:italic&gt; a non-zero &lt;jats:italic&gt;S&lt;/jats:italic&gt;-integer, and &lt;jats:italic&gt;m&lt;/jats:italic&gt; an integer &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; &lt;m:mo&gt;≥&lt;/m:mo&gt; &lt;m:mn&gt;2&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-2023-0381_eq_0497.png\" /&gt; &lt;jats:tex-math&gt;{geq 2}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. We consider the following equation &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:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mo&gt;⋆&lt;/m:mo&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&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-2023-0381_eq_0302.png\" /&gt; &lt;jats:tex-math&gt;{(star)}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;: &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: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;x&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;b&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mi&gt;y&lt;/m:mi&gt; &lt;m:mi&gt;m&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:mrow&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-2023-0381_eq_0620.png\" /&gt; &lt;jats:tex-math&gt;{f(x)=by^{m}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; in &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:mrow&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;y&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mi&gt;O&lt;/m:mi&gt; &lt;m:mi&gt;S&lt;/m:mi&gt; &lt;/m:msub&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-2023-0381_eq_0734.png\" /&gt; &lt;jats:tex-math&gt;{x,yin O_{S}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of &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;K&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;S&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;m&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-2023-0381_eq_0344.png\" /&gt; &lt;jats:tex-math&gt;{K,S,f,m}&lt;/jats:tex-math&gt; &lt;/jats:altern","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"26 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300125","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":"Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N","authors":"Deepak Kumar Mahanta, Tuhina Mukherjee, Abhishek Sarkar","doi":"10.1515/forum-2023-0407","DOIUrl":"https://doi.org/10.1515/forum-2023-0407","url":null,"abstract":"In this paper, we deal with the existence of nontrivial nonnegative solutions for a <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:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>N</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-2023-0407_eq_0313.png\" /> <jats:tex-math>{(p,N)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-Laplacian Schrödinger–Kirchhoff problem in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_forum-2023-0407_eq_0469.png\" /> <jats:tex-math>{mathbb{R}^{N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with singular exponential nonlinearity. The main features of the paper are the <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:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>N</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-2023-0407_eq_0313.png\" /> <jats:tex-math>{(p,N)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140300116","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":"Homogeneous ACM and Ulrich bundles on rational homogeneous spaces","authors":"Xinyi Fang","doi":"10.1515/forum-2023-0383","DOIUrl":"https://doi.org/10.1515/forum-2023-0383","url":null,"abstract":"In this paper, we characterize homogeneous arithmetically Cohen–Macaulay (ACM) bundles and Ulrich bundles on rational homogeneous spaces. From this result, we see that there are only finitely many irreducible homogeneous ACM bundles (up to twist) and Ulrich bundles on these varieties. Moreover, we give numerical criteria for some special irreducible homogeneous bundles to be ACM bundles.","PeriodicalId":12433,"journal":{"name":"Forum Mathematicum","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140197511","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}