Pub Date : 2026-01-23DOI: 10.1016/j.bulsci.2026.103802
Xiaojuan Li , Mingshang Hu
In this paper, we obtain a new estimate for uniform integrability under sublinear expectations. Based on this, we establish the limit theorems under nonlinear expectations dominated by sublinear expectations through tightness, where the limit distributions can be completely nonlinear. Finally, we study the limit theorem in a special case, where the limit distribution satisfies positive homogeneity.
{"title":"Limit theorems under nonlinear expectations dominated by sublinear expectations","authors":"Xiaojuan Li , Mingshang Hu","doi":"10.1016/j.bulsci.2026.103802","DOIUrl":"10.1016/j.bulsci.2026.103802","url":null,"abstract":"<div><div>In this paper, we obtain a new estimate for uniform integrability under sublinear expectations. Based on this, we establish the limit theorems under nonlinear expectations dominated by sublinear expectations through tightness, where the limit distributions can be completely nonlinear. Finally, we study the limit theorem in a special case, where the limit distribution satisfies positive homogeneity.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"209 ","pages":"Article 103802"},"PeriodicalIF":0.9,"publicationDate":"2026-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146079413","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 : 2026-01-16DOI: 10.1016/j.bulsci.2026.103795
Ashima Bansal , Supravat Sarkar , Shivam Vats
We describe when two multiprojective bundles (fibre products of projective bundles over the same base) over projective spaces are isomorphic as abstract varieties. We also describe when two relative symmetric powers of projective bundles over projective spaces are isomorphic. Finally, we describe the automorphisms of multiprojective bundles and relative symmetric powers of projective bundles over projective spaces.
{"title":"Isomorphisms and automorphisms of multiprojective bundles and symmetric powers of projective bundles","authors":"Ashima Bansal , Supravat Sarkar , Shivam Vats","doi":"10.1016/j.bulsci.2026.103795","DOIUrl":"10.1016/j.bulsci.2026.103795","url":null,"abstract":"<div><div>We describe when two multiprojective bundles (fibre products of projective bundles over the same base) over projective spaces are isomorphic as abstract varieties. We also describe when two relative symmetric powers of projective bundles over projective spaces are isomorphic. Finally, we describe the automorphisms of multiprojective bundles and relative symmetric powers of projective bundles over projective spaces.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"209 ","pages":"Article 103795"},"PeriodicalIF":0.9,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146024040","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 : 2026-01-14DOI: 10.1016/j.bulsci.2026.103793
Matthew Weaver
A classical result of Micali [17] asserts that a Noetherian local ring is regular if and only if the Rees algebra of its maximal ideal is defined by an ideal of linear forms. In this case, this defining ideal may be realized as a determinantal ideal of generic height, and so the Rees ring is easily resolved by the Eagon–Northcott complex, providing a wealth of information. If R is a non-regular local ring, it is interesting to ask how far the Rees ring of its maximal ideal strays from this form, and whether any homological data can be recovered. In this paper, we answer this question for hypersurface rings, and provide a minimal generating set for the defining ideal of the Rees ring. Furthermore, we determine the Cohen–Macaulayness of this algebra, along with several other invariants.
{"title":"Blowups of hypersurfaces","authors":"Matthew Weaver","doi":"10.1016/j.bulsci.2026.103793","DOIUrl":"10.1016/j.bulsci.2026.103793","url":null,"abstract":"<div><div>A classical result of Micali <span><span>[17]</span></span> asserts that a Noetherian local ring is regular if and only if the Rees algebra of its maximal ideal is defined by an ideal of linear forms. In this case, this defining ideal may be realized as a determinantal ideal of generic height, and so the Rees ring is easily resolved by the Eagon–Northcott complex, providing a wealth of information. If <em>R</em> is a non-regular local ring, it is interesting to ask how far the Rees ring of its maximal ideal strays from this form, and whether any homological data can be recovered. In this paper, we answer this question for hypersurface rings, and provide a minimal generating set for the defining ideal of the Rees ring. Furthermore, we determine the Cohen–Macaulayness of this algebra, along with several other invariants.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"209 ","pages":"Article 103793"},"PeriodicalIF":0.9,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146024041","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 : 2026-01-05DOI: 10.1016/j.bulsci.2025.103792
Adolfo Guillot , Luís Gustavo Mendes
The works of Brunella and Santos have singled out three special singular holomorphic foliations on projective surfaces having invariant rational nodal curves of positive self-intersection. These foliations can be described as quotients of foliations on some rational surfaces under cyclic groups of transformations of orders three, four, and six, respectively. Through an unexpected connection with the reduced Chazy IV, V and VI equations, we give explicit models for these foliations as degree-two foliations on the projective plane (in particular, we recover Pereira's model of Brunella's foliation). We describe the full groups of birational automorphisms of these quotient foliations, and, through this, produce symmetries for the reduced Chazy IV and V equations. We give another model for Brunella's very special foliation, one with only non-degenerate singularities, for which its characterizing involution is a quartic de Jonquières one, and for which its order-three symmetries are linear. Lastly, our analysis of the action of monomial transformations on linear foliations poses naturally the question of determining planar models for their quotients under the action of the standard quadratic Cremona involution; we give explicit formulas for these as well.
{"title":"Birational geometry of special quotient foliations and Chazy's equations","authors":"Adolfo Guillot , Luís Gustavo Mendes","doi":"10.1016/j.bulsci.2025.103792","DOIUrl":"10.1016/j.bulsci.2025.103792","url":null,"abstract":"<div><div>The works of Brunella and Santos have singled out three special singular holomorphic foliations on projective surfaces having invariant rational nodal curves of positive self-intersection. These foliations can be described as quotients of foliations on some rational surfaces under cyclic groups of transformations of orders three, four, and six, respectively. Through an unexpected connection with the reduced Chazy IV, V and VI equations, we give explicit models for these foliations as degree-two foliations on the projective plane (in particular, we recover Pereira's model of Brunella's foliation). We describe the full groups of birational automorphisms of these quotient foliations, and, through this, produce symmetries for the reduced Chazy IV and V equations. We give another model for Brunella's very special foliation, one with only non-degenerate singularities, for which its characterizing involution is a quartic de Jonquières one, and for which its order-three symmetries are linear. Lastly, our analysis of the action of monomial transformations on linear foliations poses naturally the question of determining planar models for their quotients under the action of the standard quadratic Cremona involution; we give explicit formulas for these as well.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"209 ","pages":"Article 103792"},"PeriodicalIF":0.9,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145915425","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 : 2026-01-05DOI: 10.1016/j.bulsci.2025.103791
Danilo Costarelli, Michele Piconi
In this paper, we establish sharp bounds for a family of Kantorovich-type neural network operators within the general frameworks of Sobolev-Orlicz and Orlicz spaces. We establish both strong (in terms of the Luxemburg norm) and weak (in terms of the modular functional) estimates, using different approaches. The strong estimates are derived for spaces generated by φ-functions that are N-functions or satisfy the -condition. Such estimates also lead to convergence results with respect to the Luxemburg norm in several instances of Orlicz spaces, including the exponential case. Meanwhile, the weak estimates are achieved under less restrictive assumptions on the involved φ-function. To obtain these results, we introduce some new tools and techniques in Orlicz spaces. Central to our approach is the Orlicz Minkowski inequality, which allows us to obtain unified strong estimates for the operators. We also present a weak (modular) version of this inequality holding under weaker conditions. Additionally, we introduce a novel notion of discrete absolute φ-moments of the hybrid type, and we employ the Hardy-Littlewood maximal operator within Orlicz spaces for the asymptotic analysis. Furthermore, we introduce the new space , which is embedded in the Sobolev-Orlicz space and modularly dense in . This allows to achieve asymptotic estimates for a wider class of φ-functions, including those that do not meet the -condition. For the extension to the whole Orlicz-setting, we generalize a Sobolev-Orlicz density result given by H. Musielak using Steklov functions, providing a modular counterpart. Moreover, we explore the relationships between weak and strong Orlicz–Lipschitz classes, corresponding to the above moduli of smoothness, providing qualitative results on the rate of convergence of the operators. Finally, a (Luxemburg norm) inverse approximation theorem in Orlicz spaces has been established, from which we deduce a characterization of the corresponding Lipschitz classes in terms of the order of convergence of the operators. The latter result shows that some of the achieved estimates are sharp.
{"title":"Strong and weak sharp bounds for neural network operators in Sobolev-Orlicz spaces and their quantitative extensions to Orlicz spaces","authors":"Danilo Costarelli, Michele Piconi","doi":"10.1016/j.bulsci.2025.103791","DOIUrl":"10.1016/j.bulsci.2025.103791","url":null,"abstract":"<div><div>In this paper, we establish sharp bounds for a family of Kantorovich-type neural network operators within the general frameworks of Sobolev-Orlicz and Orlicz spaces. We establish both strong (in terms of the Luxemburg norm) and weak (in terms of the modular functional) estimates, using different approaches. The strong estimates are derived for spaces generated by <em>φ</em>-functions that are <em>N</em>-functions or satisfy the <span><math><msup><mrow><mi>Δ</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>-condition. Such estimates also lead to convergence results with respect to the Luxemburg norm in several instances of Orlicz spaces, including the exponential case. Meanwhile, the weak estimates are achieved under less restrictive assumptions on the involved <em>φ</em>-function. To obtain these results, we introduce some new tools and techniques in Orlicz spaces. Central to our approach is the Orlicz Minkowski inequality, which allows us to obtain unified strong estimates for the operators. We also present a weak (modular) version of this inequality holding under weaker conditions. Additionally, we introduce a novel notion of discrete absolute <em>φ</em>-moments of the hybrid type, and we employ the Hardy-Littlewood maximal operator within Orlicz spaces for the asymptotic analysis. Furthermore, we introduce the new space <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>φ</mi></mrow></msup><mo>(</mo><mi>I</mi><mo>)</mo></math></span>, which is embedded in the Sobolev-Orlicz space <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>φ</mi></mrow></msup><mo>(</mo><mi>I</mi><mo>)</mo></math></span> and modularly dense in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>φ</mi></mrow></msup><mo>(</mo><mi>I</mi><mo>)</mo></math></span>. This allows to achieve asymptotic estimates for a wider class of <em>φ</em>-functions, including those that do not meet the <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-condition. For the extension to the whole Orlicz-setting, we generalize a Sobolev-Orlicz density result given by H. Musielak using Steklov functions, providing a modular counterpart. Moreover, we explore the relationships between weak and strong Orlicz–Lipschitz classes, corresponding to the above moduli of smoothness, providing qualitative results on the rate of convergence of the operators. Finally, a (Luxemburg norm) inverse approximation theorem in Orlicz spaces has been established, from which we deduce a characterization of the corresponding Lipschitz classes in terms of the order of convergence of the operators. The latter result shows that some of the achieved estimates are sharp.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"208 ","pages":"Article 103791"},"PeriodicalIF":0.9,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145926545","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 : 2025-12-30DOI: 10.1016/j.bulsci.2025.103790
Li Xiong, Zhengdong Du
In this paper, we consider the number of limit cycles of a class of planar piecewise linear systems defined in three zones separated by two parallel lines with at least one closed orbit crossing all switching lines. The system can be transformed to a Liénard-like canonical form with eleven parameters. We obtain conditions for the existence of crossing and sliding limit cycles. When the system has at most one sliding point on each switching line and the traces of the matrices of two of the subsystems are zero, we prove that it has no limit cycles. When the system has at most one sliding point on each switching line, is symmetrical with respect to the origin, and is neither of real focus-focus-real focus type, nor real focus-node (either proper or improper node)-real focus type, we prove that it has at most three limit cycles. When the traces of the matrices of the three subsystems are zero, and the determinant of the left subsystem is nonnegative, we prove that it has at most one limit cycle. For the last case, if the system has one sliding limit cycle, we show that the sliding limit cycle has six possible configurations. Moreover, the upper bounds of all of those cases can be reached. In particular, we show that the system has one sliding limit cycle even if it has no real focuses.
{"title":"Limit cycles for a class of piecewise linear systems with three zones separated by two parallel lines","authors":"Li Xiong, Zhengdong Du","doi":"10.1016/j.bulsci.2025.103790","DOIUrl":"10.1016/j.bulsci.2025.103790","url":null,"abstract":"<div><div>In this paper, we consider the number of limit cycles of a class of planar piecewise linear systems defined in three zones separated by two parallel lines with at least one closed orbit crossing all switching lines. The system can be transformed to a Liénard-like canonical form with eleven parameters. We obtain conditions for the existence of crossing and sliding limit cycles. When the system has at most one sliding point on each switching line and the traces of the matrices of two of the subsystems are zero, we prove that it has no limit cycles. When the system has at most one sliding point on each switching line, is symmetrical with respect to the origin, and is neither of real focus-focus-real focus type, nor real focus-node (either proper or improper node)-real focus type, we prove that it has at most three limit cycles. When the traces of the matrices of the three subsystems are zero, and the determinant of the left subsystem is nonnegative, we prove that it has at most one limit cycle. For the last case, if the system has one sliding limit cycle, we show that the sliding limit cycle has six possible configurations. Moreover, the upper bounds of all of those cases can be reached. In particular, we show that the system has one sliding limit cycle even if it has no real focuses.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"208 ","pages":"Article 103790"},"PeriodicalIF":0.9,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145925958","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 : 2025-12-22DOI: 10.1016/j.bulsci.2025.103789
Suman Das , Jie Huang , Antti Rasila
Let , , be the class of normalized K-quasiconformal harmonic mappings in the unit disk. We obtain Baernstein type extremal results for the analytic and co-analytic parts of functions in the geometric subclasses of . We then apply these results to obtain integral means estimates for the respective classes. Furthermore, we find the range of such that these geometric classes of harmonic quasiconformal mappings are contained in the Hardy space , thereby refining some earlier results of Nowak.
{"title":"Hardy spaces of harmonic quasiconformal mappings and Baernstein's theorem","authors":"Suman Das , Jie Huang , Antti Rasila","doi":"10.1016/j.bulsci.2025.103789","DOIUrl":"10.1016/j.bulsci.2025.103789","url":null,"abstract":"<div><div>Let <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>(</mo><mi>K</mi><mo>)</mo></math></span>, <span><math><mi>K</mi><mo>≥</mo><mn>1</mn></math></span>, be the class of normalized <em>K</em>-quasiconformal harmonic mappings in the unit disk. We obtain Baernstein type extremal results for the analytic and co-analytic parts of functions in the geometric subclasses of <span><math><msubsup><mrow><mi>S</mi></mrow><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>(</mo><mi>K</mi><mo>)</mo></math></span>. We then apply these results to obtain integral means estimates for the respective classes. Furthermore, we find the range of <span><math><mi>p</mi><mo>></mo><mn>0</mn></math></span> such that these geometric classes of harmonic quasiconformal mappings are contained in the Hardy space <span><math><msup><mrow><mi>h</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>, thereby refining some earlier results of Nowak.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"208 ","pages":"Article 103789"},"PeriodicalIF":0.9,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145925957","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 : 2025-12-18DOI: 10.1016/j.bulsci.2025.103788
A. Ballester-Bolinches , L.A. Kurdachenko , P. Pérez-Altarriba , V. Pérez-Calabuig
In this article we delve into the study of braces satisfying the maximal condition on ideals. We call them i-noetherian braces. The main goal of this article is to show a brace-theoretical analogue of a well-known result of Hall for metabelian groups: we prove that a 2-multipermutational brace is i-noetherian if, and only if, it is finitely generated as a brace. An example of an i-noetherian brace that does not satisfy the maximal condition on subbraces follows naturally from our main result.
{"title":"On the structure of braces satisfying the maximal condition on ideals","authors":"A. Ballester-Bolinches , L.A. Kurdachenko , P. Pérez-Altarriba , V. Pérez-Calabuig","doi":"10.1016/j.bulsci.2025.103788","DOIUrl":"10.1016/j.bulsci.2025.103788","url":null,"abstract":"<div><div>In this article we delve into the study of braces satisfying the maximal condition on ideals. We call them <em>i-noetherian</em> braces. The main goal of this article is to show a brace-theoretical analogue of a well-known result of Hall for metabelian groups: we prove that a 2-multipermutational brace is i-noetherian if, and only if, it is finitely generated as a brace. An example of an i-noetherian brace that does not satisfy the maximal condition on subbraces follows naturally from our main result.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"208 ","pages":"Article 103788"},"PeriodicalIF":0.9,"publicationDate":"2025-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145884364","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}
In this work, we study the nonhomogeneous Dirichlet problem for the parabolic equation involving fractional Musielak Laplacian and subcritical growth conditions. Using the modified potential well method and Galerkin's method, we establish results on the local and global existence of weak and strong solutions, as well as finite-time blow-up, depending on the initial energy level (low, critical, or high). Moreover, we explore a class of nonlocal operators including the fractional Laplacian with variable exponent, the fractional Orlicz Laplacian, the fractional double-phase operator, to highlight the broad applicability of our approach.
This study contributes to the developing theory of fractional Musielak-Sobolev spaces, a field that has received limited attention in the literature. To our knowledge, this is the first work addressing the parabolic fractional Musielak Laplacian equation.
{"title":"Global existence and finite-time blow-up of solutions for parabolic equations involving the fractional Musielak Laplacian","authors":"Rakesh Arora , Anouar Bahrouni , Nitin Kumar Maurya","doi":"10.1016/j.bulsci.2025.103787","DOIUrl":"10.1016/j.bulsci.2025.103787","url":null,"abstract":"<div><div>In this work, we study the nonhomogeneous Dirichlet problem for the parabolic equation involving fractional Musielak Laplacian and subcritical growth conditions. Using the modified potential well method and Galerkin's method, we establish results on the local and global existence of weak and strong solutions, as well as finite-time blow-up, depending on the initial energy level (low, critical, or high). Moreover, we explore a class of nonlocal operators including the fractional Laplacian with variable exponent, the fractional Orlicz Laplacian, the fractional double-phase operator, to highlight the broad applicability of our approach.</div><div>This study contributes to the developing theory of fractional Musielak-Sobolev spaces, a field that has received limited attention in the literature. To our knowledge, this is the first work addressing the parabolic fractional Musielak Laplacian equation.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"208 ","pages":"Article 103787"},"PeriodicalIF":0.9,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145840681","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 : 2025-12-10DOI: 10.1016/j.bulsci.2025.103786
Mousomi Bhakta , Alessio Fiscella , Shilpa Gupta
<div><div>In this paper, we deal with the following <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-fractional problem<span><span><span><math><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mi>u</mi><mo>+</mo><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mi>u</mi><mo>=</mo><mi>λ</mi><mi>P</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mi>θ</mi><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><msubsup><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mspace></mspace><mtext> in </mtext><mspace></mspace><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace><mtext> in </mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>∖</mo><mi>Ω</mi><mo>,</mo></math></span></span></span> where <span><math><mi>Ω</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> is a general open set, <span><math><mn>0</mn><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><mn>1</mn></math></span>, <span><math><mn>1</mn><mo><</mo><mi>q</mi><mo><</mo><mi>k</mi><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi><mo>/</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, parameter <span><math><mi>λ</mi><mo>,</mo><mspace></mspace><mi>θ</mi><mo>></mo><mn>0</mn></math></span>, <em>P</em> is a nontrivial nonnegative weight, while <span><math><msubsup><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>=</mo><mi>N</mi><mi>p</mi><mo>/</mo><mo>(</mo><mi>N</mi><mo>−</mo><mi>p</mi><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> is the critical exponent. We prove that there exists a decreasing sequence <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>j</mi></mrow></msub></math></span> such that for any <span><math><mi>j</mi><mo>∈</mo><mi>N</mi></math></span> and with <span><math><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></math></span> there exist <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span>, <span><math><msup><mrow><mi>λ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>></mo><mn>0</mn></math></span> such that above problem admits at least <em>j</em> distinct weak sol
{"title":"Critical (p,q)-fractional problems involving a sandwich type nonlinearity","authors":"Mousomi Bhakta , Alessio Fiscella , Shilpa Gupta","doi":"10.1016/j.bulsci.2025.103786","DOIUrl":"10.1016/j.bulsci.2025.103786","url":null,"abstract":"<div><div>In this paper, we deal with the following <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span>-fractional problem<span><span><span><math><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mi>u</mi><mo>+</mo><msubsup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mi>u</mi><mo>=</mo><mi>λ</mi><mi>P</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mi>θ</mi><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><msubsup><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mspace></mspace><mtext> in </mtext><mspace></mspace><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace><mtext> in </mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>∖</mo><mi>Ω</mi><mo>,</mo></math></span></span></span> where <span><math><mi>Ω</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span> is a general open set, <span><math><mn>0</mn><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><mn>1</mn></math></span>, <span><math><mn>1</mn><mo><</mo><mi>q</mi><mo><</mo><mi>k</mi><mo><</mo><mi>p</mi><mo><</mo><mi>N</mi><mo>/</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, parameter <span><math><mi>λ</mi><mo>,</mo><mspace></mspace><mi>θ</mi><mo>></mo><mn>0</mn></math></span>, <em>P</em> is a nontrivial nonnegative weight, while <span><math><msubsup><mrow><mi>p</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>=</mo><mi>N</mi><mi>p</mi><mo>/</mo><mo>(</mo><mi>N</mi><mo>−</mo><mi>p</mi><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> is the critical exponent. We prove that there exists a decreasing sequence <span><math><msub><mrow><mo>{</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>j</mi></mrow></msub></math></span> such that for any <span><math><mi>j</mi><mo>∈</mo><mi>N</mi></math></span> and with <span><math><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></math></span> there exist <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span>, <span><math><msup><mrow><mi>λ</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>></mo><mn>0</mn></math></span> such that above problem admits at least <em>j</em> distinct weak sol","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"208 ","pages":"Article 103786"},"PeriodicalIF":0.9,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145796566","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}
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