Abstract In this article, we are concerned with multiple solutions of Schrödinger-Choquard-Kirchhoff equations involving the fractional p p -Laplacian and Hardy-Littlewood-Sobolev critical exponents in R N {{mathbb{R}}}^{N} . We classify the multiplicity of the solutions in accordance with the Kirchhoff term M ( ⋅ ) Mleft(cdot ) and different ranges of q q shown in the nonlinearity f ( x , ⋅ ) fleft(x,cdot ) by means of the variational methods and Krasnoselskii’s genus theory. As an immediate consequence, some recent related results have been improved and extended.
{"title":"Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents","authors":"Xiaolu Lin, Shenzhou Zheng, Z. Feng","doi":"10.1515/ans-2022-0059","DOIUrl":"https://doi.org/10.1515/ans-2022-0059","url":null,"abstract":"Abstract In this article, we are concerned with multiple solutions of Schrödinger-Choquard-Kirchhoff equations involving the fractional p p -Laplacian and Hardy-Littlewood-Sobolev critical exponents in R N {{mathbb{R}}}^{N} . We classify the multiplicity of the solutions in accordance with the Kirchhoff term M ( ⋅ ) Mleft(cdot ) and different ranges of q q shown in the nonlinearity f ( x , ⋅ ) fleft(x,cdot ) by means of the variational methods and Krasnoselskii’s genus theory. As an immediate consequence, some recent related results have been improved and extended.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48903057","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We show that there is a one-to-one correspondence between solutions to the Poisson-landscape equations and the reduced Hartree-Fock equations in the semi-classical limit at low temperature. Moreover, we prove that the difference between the two corresponding solutions is small by providing explicit estimates.
{"title":"On an effective equation of the reduced Hartree-Fock theory","authors":"Ilias Chenn, S. Mayboroda, Wei Wang, Shiwen Zhang","doi":"10.1515/ans-2022-0070","DOIUrl":"https://doi.org/10.1515/ans-2022-0070","url":null,"abstract":"Abstract We show that there is a one-to-one correspondence between solutions to the Poisson-landscape equations and the reduced Hartree-Fock equations in the semi-classical limit at low temperature. Moreover, we prove that the difference between the two corresponding solutions is small by providing explicit estimates.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41821455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Our main purpose is to establish Gagliardo-Nirenberg-type inequalities using fractional homogeneous Sobolev spaces and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in previous studies.
{"title":"Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces","authors":"N. Dao","doi":"10.1515/ans-2022-0080","DOIUrl":"https://doi.org/10.1515/ans-2022-0080","url":null,"abstract":"Abstract Our main purpose is to establish Gagliardo-Nirenberg-type inequalities using fractional homogeneous Sobolev spaces and homogeneous Besov spaces. In particular, we extend some of the results obtained by the authors in previous studies.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"53 1-2","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41297602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yahui Jiang, Taiyong Chen, Jianjun Zhang, M. Squassina, N. Almousa
Abstract We are concerned with the following coupled nonlinear Schrödinger system: − Δ u + u + ∫ ∣ x ∣ ∞ h ( s ) s u 2 ( s ) d s + h 2 ( ∣ x ∣ ) ∣ x ∣ 2 u = ∣ u ∣ 2 p − 2 u + b ∣ v ∣ p ∣ u ∣ p − 2 u , x ∈ R 2 , − Δ v + ω v + ∫ ∣ x ∣ ∞ g ( s ) s v 2 ( s ) d s + g 2 ( ∣ x ∣ ) ∣ x ∣ 2 v = ∣ v ∣ 2 p − 2 v + b ∣ u ∣ p ∣ v ∣ p − 2 v , x ∈ R 2 , left{begin{array}{l}-Delta u+u+left(underset{| x| }{overset{infty }{displaystyle int }}frac{hleft(s)}{s}{u}^{2}left(s){rm{d}}s+frac{{h}^{2}left(| x| )}{{| x| }^{2}}right)u={| u| }^{2p-2}u+b{| v| }^{p}{| u| }^{p-2}u,hspace{1em}xin {{mathbb{R}}}^{2},hspace{1.0em} -Delta v+omega v+left(underset{| x| }{overset{infty }{displaystyle int }}frac{gleft(s)}{s}{v}^{2}left(s){rm{d}}s+frac{{g}^{2}left(| x| )}{{| x| }^{2}}right)v={| v| }^{2p-2}v+b{| u| }^{p}{| v| }^{p-2}v,hspace{1em}xin {{mathbb{R}}}^{2},hspace{1.0em}end{array}right. where ω , b > 0 omega ,bgt 0 , p > 1 pgt 1 . By virtue of the variational approach, we show the existence of nontrivial ground-state solutions depending on the parameters involved. Precisely, the aforementioned system admits a positive ground-state solution if p > 3 pgt 3 and b > 0 bgt 0 large enough or if p ∈ ( 2 , 3 ] pin left(2,3] and b > 0 bgt 0 small.
{"title":"Ground states of Schrödinger systems with the Chern-Simons gauge fields","authors":"Yahui Jiang, Taiyong Chen, Jianjun Zhang, M. Squassina, N. Almousa","doi":"10.1515/ans-2023-0086","DOIUrl":"https://doi.org/10.1515/ans-2023-0086","url":null,"abstract":"Abstract We are concerned with the following coupled nonlinear Schrödinger system: − Δ u + u + ∫ ∣ x ∣ ∞ h ( s ) s u 2 ( s ) d s + h 2 ( ∣ x ∣ ) ∣ x ∣ 2 u = ∣ u ∣ 2 p − 2 u + b ∣ v ∣ p ∣ u ∣ p − 2 u , x ∈ R 2 , − Δ v + ω v + ∫ ∣ x ∣ ∞ g ( s ) s v 2 ( s ) d s + g 2 ( ∣ x ∣ ) ∣ x ∣ 2 v = ∣ v ∣ 2 p − 2 v + b ∣ u ∣ p ∣ v ∣ p − 2 v , x ∈ R 2 , left{begin{array}{l}-Delta u+u+left(underset{| x| }{overset{infty }{displaystyle int }}frac{hleft(s)}{s}{u}^{2}left(s){rm{d}}s+frac{{h}^{2}left(| x| )}{{| x| }^{2}}right)u={| u| }^{2p-2}u+b{| v| }^{p}{| u| }^{p-2}u,hspace{1em}xin {{mathbb{R}}}^{2},hspace{1.0em} -Delta v+omega v+left(underset{| x| }{overset{infty }{displaystyle int }}frac{gleft(s)}{s}{v}^{2}left(s){rm{d}}s+frac{{g}^{2}left(| x| )}{{| x| }^{2}}right)v={| v| }^{2p-2}v+b{| u| }^{p}{| v| }^{p-2}v,hspace{1em}xin {{mathbb{R}}}^{2},hspace{1.0em}end{array}right. where ω , b > 0 omega ,bgt 0 , p > 1 pgt 1 . By virtue of the variational approach, we show the existence of nontrivial ground-state solutions depending on the parameters involved. Precisely, the aforementioned system admits a positive ground-state solution if p > 3 pgt 3 and b > 0 bgt 0 large enough or if p ∈ ( 2 , 3 ] pin left(2,3] and b > 0 bgt 0 small.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42264098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We survey some ideas regarding the application of the Aleksandrov reflection method in partial differential equation to extrinsic geometric flows of Euclidean hypersurfaces. In this survey, we mention some related and important recent developments of others on the convergence of noncontracting flows and construction and classification of ancient flows.
{"title":"Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces","authors":"B. Chow","doi":"10.1515/ans-2022-0034","DOIUrl":"https://doi.org/10.1515/ans-2022-0034","url":null,"abstract":"Abstract We survey some ideas regarding the application of the Aleksandrov reflection method in partial differential equation to extrinsic geometric flows of Euclidean hypersurfaces. In this survey, we mention some related and important recent developments of others on the convergence of noncontracting flows and construction and classification of ancient flows.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47397193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let 1 < p < ∞ 1lt plt infty and suppose that we are given a function f f defined on the leaves of a weighted tree. We would like to extend f f to a function F F defined on the entire tree, so as to minimize the weighted W 1 , p {W}^{1,p} -Sobolev norm of the extension. An easy situation is when p = 2 p=2 , where the harmonic extension operator provides such a function F F . In this note, we record our analysis of the particular case of a radially symmetric binary tree, which is a complete, finite, binary tree with weights that depend only on the distance from the root. Neither the averaging operator nor the harmonic extension operator work here in general. Nevertheless, we prove the existence of a linear extension operator whose norm is bounded by a constant depending solely on p p . This operator is a variant of the standard harmonic extension operator, and in fact, it is harmonic extension with respect to a certain Markov kernel determined by p p and by the weights.
摘要:设1 < p <∞1 lt p ltinfty，并假设给定一个函数f f定义在加权树的叶上。我们想把f扩展成定义在整棵树上的函数f，以最小化扩展的加权{W}^{1 p} -Sobolev范数。一种简单的情况是当p=2 p=2时，调和扩展算子给出了这样一个函数F F。在这篇笔记中，我们记录了我们对一个径向对称二叉树的特殊情况的分析，它是一个完全的、有限的二叉树，其权重只取决于到根的距离。一般来说，平均算子和调和扩展算子在这里都不起作用。然而，我们证明了一个线性扩展算子的存在性，其范数由一个仅依赖于p p的常数限定。这个算子是标准调和扩展算子的一个变体，事实上，它是对一个由p p和权值决定的马尔可夫核的调和扩展。
{"title":"Linear extension operators for Sobolev spaces on radially symmetric binary trees","authors":"C. Fefferman, B. Klartag","doi":"10.1515/ans-2022-0075","DOIUrl":"https://doi.org/10.1515/ans-2022-0075","url":null,"abstract":"Abstract Let 1 < p < ∞ 1lt plt infty and suppose that we are given a function f f defined on the leaves of a weighted tree. We would like to extend f f to a function F F defined on the entire tree, so as to minimize the weighted W 1 , p {W}^{1,p} -Sobolev norm of the extension. An easy situation is when p = 2 p=2 , where the harmonic extension operator provides such a function F F . In this note, we record our analysis of the particular case of a radially symmetric binary tree, which is a complete, finite, binary tree with weights that depend only on the distance from the root. Neither the averaging operator nor the harmonic extension operator work here in general. Nevertheless, we prove the existence of a linear extension operator whose norm is bounded by a constant depending solely on p p . This operator is a variant of the standard harmonic extension operator, and in fact, it is harmonic extension with respect to a certain Markov kernel determined by p p and by the weights.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46293821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we first apply the method of combining the interpolation theorem and weak-type estimate developed in Chen et al. to derive the Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using this inequality and weighted Hardy inequality, we further obtain the Stein-Weiss inequality with an extended Poisson kernel. For the extremal problem of the corresponding Stein-Weiss inequality, the presence of double-weighted exponents not being necessarily nonnegative makes it impossible to obtain the desired existence result through the usual technique of symmetrization and rearrangement. We then adopt the concentration compactness principle of double-weighted integral operator, which was first used by the authors in Chen et al. to overcome this difficulty and obtain the existence of the extremals. Finally, the regularity of the positive solution for integral system related with the extended kernel is also considered in this article. Our regularity result also avoids the nonnegativity condition of double-weighted exponents, which is a common assumption in dealing with the regularity of positive solutions of the double-weighted integral systems in the literatures.
{"title":"Integral inequalities with an extended Poisson kernel and the existence of the extremals","authors":"Chunxia Tao, Yike Wang","doi":"10.1515/ans-2023-0104","DOIUrl":"https://doi.org/10.1515/ans-2023-0104","url":null,"abstract":"Abstract In this article, we first apply the method of combining the interpolation theorem and weak-type estimate developed in Chen et al. to derive the Hardy-Littlewood-Sobolev inequality with an extended Poisson kernel. By using this inequality and weighted Hardy inequality, we further obtain the Stein-Weiss inequality with an extended Poisson kernel. For the extremal problem of the corresponding Stein-Weiss inequality, the presence of double-weighted exponents not being necessarily nonnegative makes it impossible to obtain the desired existence result through the usual technique of symmetrization and rearrangement. We then adopt the concentration compactness principle of double-weighted integral operator, which was first used by the authors in Chen et al. to overcome this difficulty and obtain the existence of the extremals. Finally, the regularity of the positive solution for integral system related with the extended kernel is also considered in this article. Our regularity result also avoids the nonnegativity condition of double-weighted exponents, which is a common assumption in dealing with the regularity of positive solutions of the double-weighted integral systems in the literatures.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45157348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We investigate the rigidity of global minimizers u ≥ 0 uge 0 of the Alt-Phillips functional involving negative power potentials ∫ Ω ( ∣ ∇ u ∣ 2 + u − γ χ { u > 0 } ) d x , γ ∈ ( 0 , 2 ) , mathop{int }limits_{Omega }(| nabla u{| }^{2}+{u}^{-gamma }{chi }_{left{ugt 0right}}){rm{d}}x,hspace{1.0em}gamma in left(0,2), when the exponent γ gamma is close to the extremes of the admissible values. In particular, we show that global minimizers in R n {{mathbb{R}}}^{n} are one-dimensional if γ gamma is close to 2 and n ≤ 7 nle 7 , or if γ gamma is close to 0 and n ≤ 4 nle 4 .
{"title":"Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents","authors":"D. De Silva, O. Savin","doi":"10.1515/ans-2022-0055","DOIUrl":"https://doi.org/10.1515/ans-2022-0055","url":null,"abstract":"Abstract We investigate the rigidity of global minimizers u ≥ 0 uge 0 of the Alt-Phillips functional involving negative power potentials ∫ Ω ( ∣ ∇ u ∣ 2 + u − γ χ { u > 0 } ) d x , γ ∈ ( 0 , 2 ) , mathop{int }limits_{Omega }(| nabla u{| }^{2}+{u}^{-gamma }{chi }_{left{ugt 0right}}){rm{d}}x,hspace{1.0em}gamma in left(0,2), when the exponent γ gamma is close to the extremes of the admissible values. In particular, we show that global minimizers in R n {{mathbb{R}}}^{n} are one-dimensional if γ gamma is close to 2 and n ≤ 7 nle 7 , or if γ gamma is close to 0 and n ≤ 4 nle 4 .","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47050501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this note, we extend the regularity theory for monotone measure-preserving maps, also known as optimal transports for the quadratic cost optimal transport problem, to the case when the support of the target measure is an arbitrary convex domain and, on the low-regularity end, between domains carrying certain invariant measures.
{"title":"Regularity properties of monotone measure-preserving maps","authors":"A. Figalli, Yash Jhaveri","doi":"10.1515/ans-2022-0057","DOIUrl":"https://doi.org/10.1515/ans-2022-0057","url":null,"abstract":"Abstract In this note, we extend the regularity theory for monotone measure-preserving maps, also known as optimal transports for the quadratic cost optimal transport problem, to the case when the support of the target measure is an arbitrary convex domain and, on the low-regularity end, between domains carrying certain invariant measures.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49584749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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