Pub Date : 2023-10-30DOI: 10.1080/17476933.2023.2272131
Qian Fu, Guantie Deng
AbstractIn this paper, we develop the theory of weighted Bergman space and obtain a general representation formula of the Bergman kernel function for the spaces on the Reinhardt domain containing the origin. As applications, we calculate the concrete forms of the Bergman kernels for some special weights on the Reinhardt domains Cn, Dn,m:={(z,w)∈Cn×Cm:‖w‖2
{"title":"The weighted reproducing kernels of the Reinhardt domain","authors":"Qian Fu, Guantie Deng","doi":"10.1080/17476933.2023.2272131","DOIUrl":"https://doi.org/10.1080/17476933.2023.2272131","url":null,"abstract":"AbstractIn this paper, we develop the theory of weighted Bergman space and obtain a general representation formula of the Bergman kernel function for the spaces on the Reinhardt domain containing the origin. As applications, we calculate the concrete forms of the Bergman kernels for some special weights on the Reinhardt domains Cn, Dn,m:={(z,w)∈Cn×Cm:‖w‖2<e−μ1‖z‖μ2} and Vη:={(z,z′,w)∈Cn×Cm×C:∑j=1neηj|w|2|zj|2+‖z′‖2<1}.Keywords: Bergman kernelweighted Bergman spaceReinhardt domainHilbert spaceAMS Subject Classifications: 32A3632A25 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThe project is supported by the National Natural Science Foundation of China (Grant no. 12071035 and 11971045).","PeriodicalId":51229,"journal":{"name":"Complex Variables and Elliptic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136069690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1080/17476933.2023.2266689
Zijian Wu, Haibo Chen
AbstractIn this paper, we consider a class of elliptic problems driven by a mixed local-nonlocal operator. By estimating the critical groups and using Morse theory, the existence of nontrivial solutions is obtained in different cases. Our results extend some classical theorems for semilinear elliptic equations to the mixed local-nonlocal setting.Keywords: Nontrivial solutionssemilinear elliptic problemscritical groupsMorse theoryAMS Subject Classifications: 35J6035J91 AcknowledgementThe authors are grateful to the anonymous referees for their useful comments and suggestions.Disclosure statementNo potential conflict of interest was reported by the author(s).Data availability statementDate sharing is not applicable to this article as no new data were created or analysed in this study.
{"title":"A class of elliptic problems driven by a mixed local-nonlocal operator","authors":"Zijian Wu, Haibo Chen","doi":"10.1080/17476933.2023.2266689","DOIUrl":"https://doi.org/10.1080/17476933.2023.2266689","url":null,"abstract":"AbstractIn this paper, we consider a class of elliptic problems driven by a mixed local-nonlocal operator. By estimating the critical groups and using Morse theory, the existence of nontrivial solutions is obtained in different cases. Our results extend some classical theorems for semilinear elliptic equations to the mixed local-nonlocal setting.Keywords: Nontrivial solutionssemilinear elliptic problemscritical groupsMorse theoryAMS Subject Classifications: 35J6035J91 AcknowledgementThe authors are grateful to the anonymous referees for their useful comments and suggestions.Disclosure statementNo potential conflict of interest was reported by the author(s).Data availability statementDate sharing is not applicable to this article as no new data were created or analysed in this study.","PeriodicalId":51229,"journal":{"name":"Complex Variables and Elliptic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135857352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-13DOI: 10.1080/17476933.2023.2266681
Zeineb Ben Yahia, Zagharide Zine El Abidine
AbstractIn this paper we investigate the existence and the asymptotic behavior of positive continuous solutions for a class of nonlinear polyharmonic boundary value problems in the punctured unit ball of Rn(n⩾3). Our arguments are based on potential theory associated to the polyharmonic operator (−Δ)m, where m is a positive integer less than n2, properties of functions in the Kato class Km,n, Karamata regular variation theory tools and the Schauder fixed point theorem.Keywords: Positive solutionsKato classKaramata classGreen functionSchauder's fixed point theoremAMS SUBJECT CLASSIFICATIONS: 35B4035B0935J3035J40 AcknowledgementThe authors express their gratitude to the referee for the careful reading of the paper and valuable recommendations.Disclosure statementNo potential conflict of interest was reported by the author(s).
{"title":"Nonlinear polyharmonic boundary value problems in the punctured unit ball","authors":"Zeineb Ben Yahia, Zagharide Zine El Abidine","doi":"10.1080/17476933.2023.2266681","DOIUrl":"https://doi.org/10.1080/17476933.2023.2266681","url":null,"abstract":"AbstractIn this paper we investigate the existence and the asymptotic behavior of positive continuous solutions for a class of nonlinear polyharmonic boundary value problems in the punctured unit ball of Rn(n⩾3). Our arguments are based on potential theory associated to the polyharmonic operator (−Δ)m, where m is a positive integer less than n2, properties of functions in the Kato class Km,n, Karamata regular variation theory tools and the Schauder fixed point theorem.Keywords: Positive solutionsKato classKaramata classGreen functionSchauder's fixed point theoremAMS SUBJECT CLASSIFICATIONS: 35B4035B0935J3035J40 AcknowledgementThe authors express their gratitude to the referee for the careful reading of the paper and valuable recommendations.Disclosure statementNo potential conflict of interest was reported by the author(s).","PeriodicalId":51229,"journal":{"name":"Complex Variables and Elliptic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135856245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-11DOI: 10.1080/17476933.2023.2260997
Seok Ban, Florian Bertrand, Amir Jaber Chehayeb, Adam Salha, Walid Tabbara
ABSTRACTWe study the higher order Kobayashi pseudometric introduced by Yu. We first obtain estimates of this pseudometric in a special pseudoconvex domain in C3. We then study the structure of the higher order extremal discs and their connection with the standard extremal discs for the Kobayashi metric.AMS SUBJECT CLASSIFICATION: 32F45 AcknowledgmentsThis work was done in the framework of the Summer Research Camp in Mathematics designed by the Department of Mathematics at the American University of Beirut (AUB) and that benefitted from a generous support from the Center for Advanced Mathematical Sciences at AUB.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingResearch of the second author was supported by the Center for Advanced Mathematical Sciences.
{"title":"On the higher order Kobayashi pseudometric","authors":"Seok Ban, Florian Bertrand, Amir Jaber Chehayeb, Adam Salha, Walid Tabbara","doi":"10.1080/17476933.2023.2260997","DOIUrl":"https://doi.org/10.1080/17476933.2023.2260997","url":null,"abstract":"ABSTRACTWe study the higher order Kobayashi pseudometric introduced by Yu. We first obtain estimates of this pseudometric in a special pseudoconvex domain in C3. We then study the structure of the higher order extremal discs and their connection with the standard extremal discs for the Kobayashi metric.AMS SUBJECT CLASSIFICATION: 32F45 AcknowledgmentsThis work was done in the framework of the Summer Research Camp in Mathematics designed by the Department of Mathematics at the American University of Beirut (AUB) and that benefitted from a generous support from the Center for Advanced Mathematical Sciences at AUB.Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingResearch of the second author was supported by the Center for Advanced Mathematical Sciences.","PeriodicalId":51229,"journal":{"name":"Complex Variables and Elliptic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136211872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-08DOI: 10.1080/17476933.2023.2260988
Zixia Yuan, Zimin Tang
In this paper we consider two classes of nonlinear partial differential equations with the fractional Laplacian, namely (−Δ)α2(um)=u|u|q−1+w(x),x∈RN,1≤mα and ∂ku∂tk+(−Δ)α2u=uq,(x,t)∈RN×(0,+∞),k≥1, 0<α≤2, N>α. Solutions defined for all x∈RN of the first equation are referred to as entire solutions, while solutions defined for all (x,t)∈RN×[0,+∞) of the second equation are referred to as global solutions. Several existence and nonexistence theorems are established over different ranges of q, and thus the respective relations between the existence, nonexistence of solutions for these equations and the index q in the nonlinear terms are obtained. It is illustrated that our results are sharp in cases of m = 1 and k = 1 respectively. In addition, we prove the positivity, symmetry and odevity of solutions we constructed for the first equation with m = 1 associated with the inhomogeneous term w.KEYWORDS: Fractional Laplaciancritical exponentexistencenonexistenceAMS SUBJECT CLASSIFICATIONS: 35B0835B3335R11 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was partially supported by the Special Project for Local Science and Technology Development Guided by the Central Government of Sichuan Province [grant number 2021ZYD0014].
{"title":"Sharp critical exponents for nonlinear equations with the fractional Laplacian","authors":"Zixia Yuan, Zimin Tang","doi":"10.1080/17476933.2023.2260988","DOIUrl":"https://doi.org/10.1080/17476933.2023.2260988","url":null,"abstract":"In this paper we consider two classes of nonlinear partial differential equations with the fractional Laplacian, namely (−Δ)α2(um)=u|u|q−1+w(x),x∈RN,1≤m<q, 0<α≤2, N>α and ∂ku∂tk+(−Δ)α2u=uq,(x,t)∈RN×(0,+∞),k≥1, 0<α≤2, N>α. Solutions defined for all x∈RN of the first equation are referred to as entire solutions, while solutions defined for all (x,t)∈RN×[0,+∞) of the second equation are referred to as global solutions. Several existence and nonexistence theorems are established over different ranges of q, and thus the respective relations between the existence, nonexistence of solutions for these equations and the index q in the nonlinear terms are obtained. It is illustrated that our results are sharp in cases of m = 1 and k = 1 respectively. In addition, we prove the positivity, symmetry and odevity of solutions we constructed for the first equation with m = 1 associated with the inhomogeneous term w.KEYWORDS: Fractional Laplaciancritical exponentexistencenonexistenceAMS SUBJECT CLASSIFICATIONS: 35B0835B3335R11 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was partially supported by the Special Project for Local Science and Technology Development Guided by the Central Government of Sichuan Province [grant number 2021ZYD0014].","PeriodicalId":51229,"journal":{"name":"Complex Variables and Elliptic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135198873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1080/17476933.2023.2260995
Liping Wang, Liping Luo, Ying Li, Xin Jiang
ABSTRACTFirstly, the definition of p order homogeneous weighted right monogenic polynomials is given, and the hypercomplex variables are introduced in order to construct a basis of all homogeneous weighted right monogenic polynomials of degree p, then the second Taylor expansion of the weighted right monogenic functions is obtained. Secondly, the weighted left monogenic functions are constructed from continuous functions in different regions, and corresponding Taylor expansions are given. Finally, on the basis of the previous conclusions, the Laurent expansion and residue theorem of the weighted left monogenic functions are proved.KEYWORDS: Weighted monogenic functionsp order homogeneous weighted right monogenic polynomialshypercomplex variablesLaurent expansionresidue theoremAMS SUBJECT CLASSIFICATIONS: 30B1030G3532A05 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by the Natural Science Foundation of Hebei Province [grant numbers A2020205008 and A2015205012], Key Foundation of Hebei Normal University [grant number L2021Z01] and the National Natural Science Foundation of China [grant numbers 11401162, 11871191, and 11571089].
{"title":"The Laurent expansion and residue theorem of weighted monogenic functions","authors":"Liping Wang, Liping Luo, Ying Li, Xin Jiang","doi":"10.1080/17476933.2023.2260995","DOIUrl":"https://doi.org/10.1080/17476933.2023.2260995","url":null,"abstract":"ABSTRACTFirstly, the definition of p order homogeneous weighted right monogenic polynomials is given, and the hypercomplex variables are introduced in order to construct a basis of all homogeneous weighted right monogenic polynomials of degree p, then the second Taylor expansion of the weighted right monogenic functions is obtained. Secondly, the weighted left monogenic functions are constructed from continuous functions in different regions, and corresponding Taylor expansions are given. Finally, on the basis of the previous conclusions, the Laurent expansion and residue theorem of the weighted left monogenic functions are proved.KEYWORDS: Weighted monogenic functionsp order homogeneous weighted right monogenic polynomialshypercomplex variablesLaurent expansionresidue theoremAMS SUBJECT CLASSIFICATIONS: 30B1030G3532A05 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThis work was supported by the Natural Science Foundation of Hebei Province [grant numbers A2020205008 and A2015205012], Key Foundation of Hebei Normal University [grant number L2021Z01] and the National Natural Science Foundation of China [grant numbers 11401162, 11871191, and 11571089].","PeriodicalId":51229,"journal":{"name":"Complex Variables and Elliptic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135458663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-28DOI: 10.1080/17476933.2023.2261004
Nguyen Van Thin, Pham Thi Thuy, Trinh Thi Diep Linh
AbstractIn this paper, we study the existence of weak solution to (p,q)-fractional Choquard equation in RN as follows Lpsu+Lqsu+V(x)(|u|p−2u+|u|q−2u)=(1|x|μ∗F(u))f(u), where 2≤Ns=p
{"title":"Existence of solution for the ( <i>p, q</i> )-fractional Laplacian equation with nonlocal Choquard reaction and exponential growth","authors":"Nguyen Van Thin, Pham Thi Thuy, Trinh Thi Diep Linh","doi":"10.1080/17476933.2023.2261004","DOIUrl":"https://doi.org/10.1080/17476933.2023.2261004","url":null,"abstract":"AbstractIn this paper, we study the existence of weak solution to (p,q)-fractional Choquard equation in RN as follows Lpsu+Lqsu+V(x)(|u|p−2u+|u|q−2u)=(1|x|μ∗F(u))f(u), where 2≤Ns=p<q, 0≤μ<N, f has exponential growth. The function f and V are continuous which satisfy some suitable assumptions. In our best knowledge, sofar there is not any work about existence of weak solution to (p,q)-fractional Choquard equation with exponential growth. Our results are even new in the case (N,q)-Laplace equation which are complement the results of Fiscella and Pucci [(p,N) equations with critical exponential nonlinearities in RN. J Math Anal Appl. 2021;501(1), Article 123379].Keywords: Integrodifferential operatorsTrudinger–Moser inequalityfractional (p,q)-Laplacianmountain pass theoremvariational methodAMS Subject Classifications: Primary 35A1535A2335J3535J6035R11 Disclosure statementNo potential conflict of interest was reported by the author(s).Additional informationFundingThe author wishes to express thanks to the referee and editorial board for reading the manuscript very carefully and making some valuable suggestions and comments towards the improvement of the paper. The first author would like to thank Prof. Claudianor O. Alves for useful comments to improve the first version of this paper. The research results are supported by Ministry of Education and Training of Vietnam under project with the name ‘Some properties about solutions to differential equations, fractional partial differential equations’ and grant number B2023-TNA-14.","PeriodicalId":51229,"journal":{"name":"Complex Variables and Elliptic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135425209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-25DOI: 10.1080/17476933.2023.2260991
Moussa Faress, Said Fahlaoui
AbstractIn this paper, after recalling the main properties of the voice transform, we prove its inversion formula, then we present some qualitative uncertainty principles associated with this transform. Finally, we find a solution to an interpolation problem.Keywords: Representationvoice transformuncertainty principlesinterpolationAMS Subject Classifications: 43-XX43A6544-XX42A3805C62 AcknowledgmentsThe authors wish to express their thanks to the referee for his/her valuable suggestions and comments.Disclosure statementNo potential conflict of interest was reported by the author(s).
{"title":"Some uncertainty principles for the continuous voice transform","authors":"Moussa Faress, Said Fahlaoui","doi":"10.1080/17476933.2023.2260991","DOIUrl":"https://doi.org/10.1080/17476933.2023.2260991","url":null,"abstract":"AbstractIn this paper, after recalling the main properties of the voice transform, we prove its inversion formula, then we present some qualitative uncertainty principles associated with this transform. Finally, we find a solution to an interpolation problem.Keywords: Representationvoice transformuncertainty principlesinterpolationAMS Subject Classifications: 43-XX43A6544-XX42A3805C62 AcknowledgmentsThe authors wish to express their thanks to the referee for his/her valuable suggestions and comments.Disclosure statementNo potential conflict of interest was reported by the author(s).","PeriodicalId":51229,"journal":{"name":"Complex Variables and Elliptic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135864067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-12DOI: 10.1080/17476933.2023.2250736
Jin-Fu Yang, Jia-Feng Zhang, Wen-Min Li, Qin Qin
{"title":"Existence of infinitely many solutions for a class of Schrödinger–Poisson equations without any growth and Ambrosetti–Rabinowitz conditions in RN","authors":"Jin-Fu Yang, Jia-Feng Zhang, Wen-Min Li, Qin Qin","doi":"10.1080/17476933.2023.2250736","DOIUrl":"https://doi.org/10.1080/17476933.2023.2250736","url":null,"abstract":"","PeriodicalId":51229,"journal":{"name":"Complex Variables and Elliptic Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135878494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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