Electronic Journal of Qualitative Theory of Differential Equations最新文献

{"title":"Positive solutions to three classes of non-local fourth-order problems with derivative-dependent nonlinearities","authors":"Guowei Zhang","doi":"10.14232/ejqtde.2022.1.11","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.11","url":null,"abstract":"In the article, we investigate three classes of fourth-order boundary value problems with dependence on all derivatives in nonlinearities under the boundary conditions involving Stieltjes integrals. A Gronwall-type inequality is employed to get an a priori bound on the third-order derivative term, and the theory of fixed-point index is used on suitable open sets to obtain the existence of positive solutions. The nonlinearities have quadratic growth in the third-order derivative term. Previous results in the literature are not applicable in our case, as shown by our examples.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66583711","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}

{"title":"Global existence and blow-up for semilinear parabolic equation with critical exponent in \u0000 \u0000 \u0000 R\u0000 \u0000 N","authors":"Fei Fang, Binlin Zhang","doi":"10.14232/ejqtde.2022.1.3","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.3","url":null,"abstract":"In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in R N . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the L 2 norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584760","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}

{"title":"Fixed-time and state-dependent time discontinuities in the theory of Stieltjes differential equations","authors":"B. Satco","doi":"10.14232/ejqtde.2022.1.28","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.28","url":null,"abstract":"In the present paper, we are concerned with a very general problem, namely the Stieltjes differential Cauchy problem involving state-dependent discontinuities. Given that the theory of Stieltjes differential equations covers the framework of impulsive problems with fixed-time impulses, in the present work we generalize this setting by allowing the occurrence of fixed-time impulses, as well as the occurrence of state-dependent impulses. Along with an existence result obtained under an overarching set of assumptions involving Stieltjes integrals, it is showed that a least and a greatest solution can be found.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584878","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}

{"title":"Asymptotic behavior of solutions to the multidimensional semidiscrete diffusion equation","authors":"A. Slavík","doi":"10.14232/ejqtde.2022.1.9","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.9","url":null,"abstract":"We study the asymptotic behavior of solutions to the multidimensional diffusion (heat) equation with continuous time and discrete space. We focus on initial-value problems with bounded initial data, and provide sufficient conditions for the existence of pointwise and uniform limits of solutions.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585097","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}

{"title":"Uniqueness criteria for ordinary differential equations with a generalized transversality condition at the initial condition","authors":"J. Cid, Rodrigo López Pouso, Jorge Rodríguez–López","doi":"10.14232/ejqtde.2022.1.6","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.6","url":null,"abstract":"In this paper, we present some uniqueness results for systems of ordinary differential equations. All of them are linked by a weak transversality condition at the initial condition, which generalizes those in the previous literature. Several examples are also provided to illustrate our results.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585248","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}

{"title":" C 1 , γ ","authors":"Duan Wu, P. Niu","doi":"10.14232/ejqtde.2022.1.29","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.29","url":null,"abstract":"In this note, we prove the boundary and global C 1 , γ regularity for viscosity solutions of fully nonlinear uniformly elliptic equations on a convex polyhedron by perturbation and iteration techniques.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584590","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}

{"title":"Sobolev inequality with non-uniformly degenerating gradient","authors":"F. Mamedov, S. Monsurrò","doi":"10.14232/ejqtde.2022.1.24","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.24","url":null,"abstract":"<jats:p>In this paper we prove the following weighted Sobolev inequality in a bounded domain <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> </mml:math>, of a homogeneous space <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math>, under suitable compatibility conditions on the positive weight functions <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ω<!-- ω --></mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math> and on the quasi-metric <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ρ<!-- ρ --></mml:mi> </mml:math>, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">(</mml:mo> </mml:mrow> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> </mml:msub> <mml:mo fence=\"false\" stretchy=\"false\">|</mml:mo> <mml:mi>f</mml:mi> <mml:msup> <mml:mo fence=\"false\" stretchy=\"false\">|</mml:mo> <mml:mi>q</mml:mi> </mml:msup> <mml:mi>v</mml:mi> <mml:mspace width=\"thinmathspace\" /> <mml:mi>w</mml:mi> <mml:mi>d</mml:mi> <mml:mi>z</mml:mi> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.623em\" minsize=\"1.623em\">)</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>q</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> ","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584378","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}

{"title":"The existence of ground state solutions for semi-linear degenerate Schrödinger equations with steep potential well","authors":"Ling Ran, Shang-Jie Chen, Lin Li","doi":"10.14232/ejqtde.2022.1.30","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.30","url":null,"abstract":"<jats:p>In this article, we study the following degenerated Schrödinger equations: <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mtable columnalign=\"right left right left right left right left right left right left\" rowspacing=\"3pt\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\"> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=\"left left\" rowspacing=\".2em\" columnspacing=\"1em\" displaystyle=\"false\"> <mml:mtr> <mml:mtd> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>γ<!-- γ --></mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mi>V</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mtd> <mml:mtd> <mml:mtext>in</mml:mtext> <mml:mtext> </mml:mtext> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>u</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>λ<!-- λ --></mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mtext> </mml:mtext> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\" /> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:math> where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> ","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66584813","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}

{"title":"On the existence of periodic solutions to second order Hamiltonian systems","authors":"Xiao-Feng Ke, Jia‐Feng Liao","doi":"10.14232/ejqtde.2022.1.36","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.36","url":null,"abstract":"In this paper, the existence of periodic solutions to the second order Hamiltonian systems is investigated. By introducing a new growth condition which generalizes the Ambrosetti–Rabinowitz condition, we prove a existence result of nontrivial $T$-periodic solution via the variational methods. Our result is new because it can deal with not only the superquadratic case, but also the anisotropic case which allows the potential to be superquadratic growth in only one direction and asymptotically quadratic growth in other directions.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585274","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}

{"title":"Strong solutions for singular Dirichlet elliptic problems","authors":"T. Godoy","doi":"10.14232/ejqtde.2022.1.40","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.40","url":null,"abstract":"<jats:p>We prove an existence result for strong solutions <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>u</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of singular semilinear elliptic problems of the form <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> </mml:math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> </mml:math> on <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> </mml:math> where <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>1</mml:mn> <mml:mo><</mml:mo> <mml:mi>q</mml:mi> <mml:mo><</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo>,</mml:mo> </mml:math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi> </mml:math> is a bounded domain in <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math> with <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> boundary, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>0</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585532","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}