. This paper, we study the multiplicity of solutions for the fractional Schr¨odinger equation with s ∈ ( 0,1 ) , N ≥ 3, p ∈ ( 1, 2 N N − 2 s − 1 ) and lim | y |→ + ∞ V ( y ) > 0. By assuming suitable decay property of the radial potential V ( y ) = V ( | y | ) , we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides. Hence, besides the length of each polygonal, we must consider one more parameter, that is the height of the podetium, simultaneously. Another difficulty lies in the non-local property of the operator ( − ∆ ) s and the algebraic decay involving the approximation solutions make the estimates become more subtle.
． 本文研究了s∈(0,1)，N≥3,p∈(1,2 N N−2 s−1)，lim | y |→+∞V (y) > 0的分数阶Schr¨odinger方程解的多重性。通过假设径向势V (y) = V (| y |)具有合适的衰减性质，构造了另一类集中于两个边长无限大的类似等边多边形无穷顶点处的解。因此，除了每个多边形的长度外，我们还必须同时考虑另一个参数，即足架的高度。另一个困难在于算子(−∆)s的非局部性质，以及涉及近似解的代数衰减使估计变得更加微妙。
{"title":"Infinitely Many Solutions for the Fractional Nonlinear Schrödinger Equations of a New Type","authors":"Qing Guo null, Lixiu Duan","doi":"10.4208/jpde.v35.n3.5","DOIUrl":"https://doi.org/10.4208/jpde.v35.n3.5","url":null,"abstract":". This paper, we study the multiplicity of solutions for the fractional Schr¨odinger equation with s ∈ ( 0,1 ) , N ≥ 3, p ∈ ( 1, 2 N N − 2 s − 1 ) and lim | y |→ + ∞ V ( y ) > 0. By assuming suitable decay property of the radial potential V ( y ) = V ( | y | ) , we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides. Hence, besides the length of each polygonal, we must consider one more parameter, that is the height of the podetium, simultaneously. Another difficulty lies in the non-local property of the operator ( − ∆ ) s and the algebraic decay involving the approximation solutions make the estimates become more subtle.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83973831","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}
. Let ( X , d , µ ) be a metric space with a Borel-measure µ , suppose µ satisfies the Ahlfors-regular condition, i.e. where b 1 , b 2 are two positive constants and s is the volume growth exponent. In this paper, we mainly study two things, one is to consider the best constant of the Moser-Trudinger inequality on such metric space under the condition that s is not less than 2. The other is to study the generalized Moser-Trudinger inequality with a singular weight.
{"title":"A Singular Moser-Trudinger Inequality on Metric Measure Space","authors":"Yaoting Gui","doi":"10.4208/jpde.v35.n4.3","DOIUrl":"https://doi.org/10.4208/jpde.v35.n4.3","url":null,"abstract":". Let ( X , d , µ ) be a metric space with a Borel-measure µ , suppose µ satisfies the Ahlfors-regular condition, i.e. where b 1 , b 2 are two positive constants and s is the volume growth exponent. In this paper, we mainly study two things, one is to consider the best constant of the Moser-Trudinger inequality on such metric space under the condition that s is not less than 2. The other is to study the generalized Moser-Trudinger inequality with a singular weight.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86043312","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}
. In this paper, we study existence of solutions of mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs. Our method is based on calculus of variations, which was built on connected finite graphs by Grigor’yan, Lin and Yang.
{"title":"Mean Field Equations for the Equilibrium Turbulence and Toda Systems on Connected Finite Graphs","authors":"Xiao-Dan Zhu","doi":"10.4208/jpde.v35.n3.1","DOIUrl":"https://doi.org/10.4208/jpde.v35.n3.1","url":null,"abstract":". In this paper, we study existence of solutions of mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs. Our method is based on calculus of variations, which was built on connected finite graphs by Grigor’yan, Lin and Yang.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74446690","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}
. In this paper, we consider the two-dimensional (2D) Prandtl-Hartmann equations on the half plane and prove the global existence and uniqueness of solutions to 2D Prandtl-Hartmann equations by using the classical energy methods in analytic framework. We prove that the lifespan of the solutions to 2D Prandtl-Hartmann equations can be extended up to T ε (see Theorem 2.1) when the strength of the perturbation is of the order of ε . The difficulty of solving the Prandtl-Hartmann equations in the analytic framework is the loss of x -derivative in the term v ∂ y u . To overcome this difficulty, we introduce the Gaussian weighted Poincar´ e inequality (see Lemma 2.3). Com-pared to the existence and uniqueness of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays a key role, which is not needed for the 2D Prandtl-Hartmann equations in analytic framework. Besides, the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role, which is not needed for the 2D Prandtl-Hartmann equations in analytic framework, either.
{"title":"Global Well-Posedness of Solutions to 2D Prandtl-Hartmann Equations in Analytic Framework","authors":"Xiaolei Dong null, Yuming Qin","doi":"10.4208/jpde.v35.n3.7","DOIUrl":"https://doi.org/10.4208/jpde.v35.n3.7","url":null,"abstract":". In this paper, we consider the two-dimensional (2D) Prandtl-Hartmann equations on the half plane and prove the global existence and uniqueness of solutions to 2D Prandtl-Hartmann equations by using the classical energy methods in analytic framework. We prove that the lifespan of the solutions to 2D Prandtl-Hartmann equations can be extended up to T ε (see Theorem 2.1) when the strength of the perturbation is of the order of ε . The difficulty of solving the Prandtl-Hartmann equations in the analytic framework is the loss of x -derivative in the term v ∂ y u . To overcome this difficulty, we introduce the Gaussian weighted Poincar´ e inequality (see Lemma 2.3). Com-pared to the existence and uniqueness of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays a key role, which is not needed for the 2D Prandtl-Hartmann equations in analytic framework. Besides, the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role, which is not needed for the 2D Prandtl-Hartmann equations in analytic framework, either.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74810291","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}
. In this paper, we consider the Cauchy problem of 3D tropical climate model with zero thermal diffusion. Firstly, we establish the global regularity for this system with fractional diffusion α = β = 5/4. Secondly, by adding only a damp term, we obtain the global well-posedness for small initial data.
{"title":"Global Well-Posedness for the 3D Tropical Climate Model without Thermal Diffusion","authors":"Yanghai Yu, Jinlu Li null, Xiuwei Yin","doi":"10.4208/jpde.v35.n4.4","DOIUrl":"https://doi.org/10.4208/jpde.v35.n4.4","url":null,"abstract":". In this paper, we consider the Cauchy problem of 3D tropical climate model with zero thermal diffusion. Firstly, we establish the global regularity for this system with fractional diffusion α = β = 5/4. Secondly, by adding only a damp term, we obtain the global well-posedness for small initial data.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89539675","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}
. The purpose of this paper is to investigate the nonexistence of positive solutions of the following doubly nonlinear degenerate parabolic equations: where Ω is a Carnot-Carath´eodory metric ball in R 2 n + 1 generated by Greiner vector fields, V ∈ L 1 loc ( Ω ) , k ∈ N , m ∈ R , 1 < p < 2 n + 2 k and m + p − 2 > 0. The better lower bound p ∗ for m + p is found and the nonexistence results are proved for p ∗ 6 m + p < 3.
． 本文的目的是研究以下双非线性退化抛物方程的正解的不存在性:其中Ω是由Greiner向量场生成的r2n + 1中的Carnot-Carath ' eodory公制球，V∈l1loc (Ω)， k∈n, m∈R, 1 < p < 2n + 2k, m + p−2 > 0。找到了m + p的较好的下界p∗，并证明了p∗6 m + p < 3时的不存在性结果。
{"title":"Doubly Nonlinear Degenerate Parabolic Equations with a Singular Potential for Greiner Vector Fields","authors":"Junqiang Han","doi":"10.4208/jpde.v35.n4.1","DOIUrl":"https://doi.org/10.4208/jpde.v35.n4.1","url":null,"abstract":". The purpose of this paper is to investigate the nonexistence of positive solutions of the following doubly nonlinear degenerate parabolic equations: where Ω is a Carnot-Carath´eodory metric ball in R 2 n + 1 generated by Greiner vector fields, V ∈ L 1 loc ( Ω ) , k ∈ N , m ∈ R , 1 < p < 2 n + 2 k and m + p − 2 > 0. The better lower bound p ∗ for m + p is found and the nonexistence results are proved for p ∗ 6 m + p < 3.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72821918","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}
. Denote v ≥ ψ , a.e. and v − θ p Ω ) o where ψ is any function in Ω ⊂ R N , N ≥ 2, with values in R ∪{± ∞ } and θ is a measurable function. This paper deals with global integrability for u ∈K ψ , θ such that with |A ( x , ξ ) |≈| ξ | p − 1 , 1 < p < N . Some global integrability results are obtained.
。表示v≥ψ， a.e.和v - θ p Ω) o，其中ψ是Ω∧R N, N≥2中的任意函数，其值在R∪{±∞}中，θ是可测函数。本文研究了u∈K ψ， θ的整体可积性，使得|A (x， ξ) |≈| ξ | p−1,1 < p < N。得到了一些全局可积性的结果。
{"title":"Global Integrability for Solutions to Obstacle Problems","authors":"Yanan Shan null, Hongya Gao","doi":"10.4208/jpde.v35.n4.2","DOIUrl":"https://doi.org/10.4208/jpde.v35.n4.2","url":null,"abstract":". Denote v ≥ ψ , a.e. and v − θ p Ω ) o where ψ is any function in Ω ⊂ R N , N ≥ 2, with values in R ∪{± ∞ } and θ is a measurable function. This paper deals with global integrability for u ∈K ψ , θ such that with |A ( x , ξ ) |≈| ξ | p − 1 , 1 < p < N . Some global integrability results are obtained.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81596895","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 Obstacle Problem For Nonlinear Degenerate Elliptic Equations with Variable Exponents and L$^1$-Data","authors":"H. Khelifi","doi":"10.4208/jpde.v35.n2.1","DOIUrl":"https://doi.org/10.4208/jpde.v35.n2.1","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91218907","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 for the Porous Media Equations with Nonlinear Norm-type Sources","authors":"Yingzhen Xue","doi":"10.4208/jpde.v35.n3.4","DOIUrl":"https://doi.org/10.4208/jpde.v35.n3.4","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91302621","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}
In this paper, we prove the symmetry of the solution to overdetermined problem for the equation σk(D 2 u − uI) = C n in hyperbolic space. Our approach is based on establishing a RellichPohozaev type identity and using a P function. Our result generalizes the overdetermined problem for Hessian equation in Euclidean space.
{"title":"Serrin-Type Overdetermined Problem in $mathbb H^n$","authors":"Zhenghuan Gao, Xiaohan Jia, Jingquan Yan","doi":"10.4208/jpde.v36.n1.7","DOIUrl":"https://doi.org/10.4208/jpde.v36.n1.7","url":null,"abstract":"In this paper, we prove the symmetry of the solution to overdetermined problem for the equation σk(D 2 u − uI) = C n in hyperbolic space. Our approach is based on establishing a RellichPohozaev type identity and using a P function. Our result generalizes the overdetermined problem for Hessian equation in Euclidean space.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75128768","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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