Advances in Nonlinear Analysis最新文献

{"title":"k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities","authors":"Xingyue He, Chenghua Gao, Jingjing Wang","doi":"10.1515/anona-2023-0136","DOIUrl":"https://doi.org/10.1515/anona-2023-0136","url":null,"abstract":"\u0000 <jats:p>In this article, our main aim is to investigate the existence of radial <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0136_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>k</m:mi>\u0000 </m:math>\u0000 <jats:tex-math>k</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>-convex solutions for the following Dirichlet system with <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0136_eq_002.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:mi>k</m:mi>\u0000 </m:math>\u0000 <jats:tex-math>k</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula>-Hessian operators: <jats:disp-formula id=\"j_anona-2023-0136_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0136_eq_003.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\u0000 <m:mfenced open=\"{\" close=\"\">\u0000 <m:mrow>\u0000 <m:mtable displaystyle=\"true\">\u0000 <m:mtr>\u0000 <m:mtd columnalign=\"left\">\u0000 <m:msub>\u0000 <m:mrow>\u0000 <m:mi>S</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>k</m:mi>\u0000 </m:mrow>\u0000 </m:msub>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi>D</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mn>2</m:mn>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 ","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140519518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a nonlinear Robin problem with an absorption term on the boundary and L\u0000 1 data","authors":"Francesco Della Pietra, Francescantonio Oliva, Sergio Segura de León","doi":"10.1515/anona-2023-0118","DOIUrl":"https://doi.org/10.1515/anona-2023-0118","url":null,"abstract":"\u0000 <jats:p>We deal with existence and uniqueness of nonnegative solutions to: <jats:disp-formula id=\"j_anona-2023-0118_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0118_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\u0000 <m:mfenced open=\"{\" close=\"\">\u0000 <m:mrow>\u0000 <m:mtable displaystyle=\"true\">\u0000 <m:mtr>\u0000 <m:mtd columnalign=\"left\">\u0000 <m:mo>−</m:mo>\u0000 <m:mi mathvariant=\"normal\">Δ</m:mi>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>=</m:mo>\u0000 <m:mi>f</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>x</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:mo>,</m:mo>\u0000 <m:mspace width=\"1.0em\" />\u0000 </m:mtd>\u0000 <m:mtd columnalign=\"left\">\u0000 <m:mstyle>\u0000 <m:mspace width=\"0.1em\" />\u0000 <m:mtext>in</m:mtext>\u0000 <m:mspace width=\"0.1em\" />\u0000 </m:mstyle>\u0000 <m:mspace width=\"0.33em\" />\u0000 <m:mi mathvariant=\"normal\">Ω</m:mi>\u0000 <m:mo>,</m:mo>\u0000 </m:mtd>\u0000 </m:mtr>\u0000 <m:mtr>\u0000 <m:mtd columnalign=\"left\">\u0000 <m:mfrac>\u0000 <m:mrow>\u0000 <m:mo>∂</m:mo>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mo>∂</m:mo>\u0000 <m:mi>ν</m:mi>\u0000 </m:mrow>\u0000 </m:mfrac>\u0000 <m:mo>+</m:mo>\u0000 ","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140516567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence and uniqueness of solution for a singular elliptic differential equation","authors":"Shanshan Gu, Bianxia Yang, Wenrui Shao","doi":"10.1515/anona-2023-0126","DOIUrl":"https://doi.org/10.1515/anona-2023-0126","url":null,"abstract":"\u0000 <jats:p>In this article, we are concerned about the existence, uniqueness, and nonexistence of the positive solution for: <jats:disp-formula id=\"j_anona-2023-0126_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0126_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\u0000 <m:mfenced open=\"{\" close=\"\">\u0000 <m:mrow>\u0000 <m:mtable displaystyle=\"true\">\u0000 <m:mtr>\u0000 <m:mtd columnalign=\"left\">\u0000 <m:mo>−</m:mo>\u0000 <m:mi mathvariant=\"normal\">Δ</m:mi>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>−</m:mo>\u0000 <m:mfrac>\u0000 <m:mrow>\u0000 <m:mn>1</m:mn>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mn>2</m:mn>\u0000 </m:mrow>\u0000 </m:mfrac>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>x</m:mi>\u0000 <m:mo>⋅</m:mo>\u0000 <m:mrow>\u0000 <m:mo>∇</m:mo>\u0000 </m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:mo>=</m:mo>\u0000 <m:mi>μ</m:mi>\u0000 <m:mi>h</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>x</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 ","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140524673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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 decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3","authors":"Leilei Tong","doi":"10.1515/anona-2023-0131","DOIUrl":"https://doi.org/10.1515/anona-2023-0131","url":null,"abstract":"\u0000 <jats:p>The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0131_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:msub>\u0000 <m:mrow>\u0000 <m:mi>ρ</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>s</m:mi>\u0000 </m:mrow>\u0000 </m:msub>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>x</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:mo>,</m:mo>\u0000 <m:mn>0</m:mn>\u0000 <m:mo>,</m:mo>\u0000 <m:mn>0</m:mn>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mi>T</m:mi>\u0000 </m:mrow>\u0000 </m:msup>\u0000 </m:math>\u0000 <jats:tex-math>{left({rho }_{s}left(x),0,0)}^{T}</jats:tex-math>\u0000 </jats:alternatives>\u0000 </jats:inline-formula> is in a small neighborhood of the constant state <jats:inline-formula>\u0000 <jats:alternatives>\u0000 <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0131_eq_002.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\">\u0000 ","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140519507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence of normalized peak solutions for a coupled nonlinear Schrödinger system","authors":"Jing Yang","doi":"10.1515/anona-2023-0113","DOIUrl":"https://doi.org/10.1515/anona-2023-0113","url":null,"abstract":"\u0000 <jats:p>In this article, we study the following nonlinear Schrödinger system <jats:disp-formula id=\"j_anona-2023-0113_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0113_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\u0000 <m:mfenced open=\"{\" close=\"\">\u0000 <m:mrow>\u0000 <m:mtable displaystyle=\"true\">\u0000 <m:mtr>\u0000 <m:mtd columnalign=\"left\">\u0000 <m:mo>−</m:mo>\u0000 <m:mi mathvariant=\"normal\">Δ</m:mi>\u0000 <m:msub>\u0000 <m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mn>1</m:mn>\u0000 </m:mrow>\u0000 </m:msub>\u0000 <m:mo>+</m:mo>\u0000 <m:msub>\u0000 <m:mrow>\u0000 <m:mi>V</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mn>1</m:mn>\u0000 </m:mrow>\u0000 </m:msub>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>x</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:msub>\u0000 <m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mn>1</m:mn>\u0000 </m:mrow>\u0000 </m:msub>\u0000 <m:mo>=</m:mo>\u0000 <m:mi>α</m:mi>\u0000 <m:msub>\u0000 <m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140518379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces","authors":"Xiangyun Xie, Yu Liu, Pengtao Li, Jizheng Huang","doi":"10.1515/anona-2023-0119","DOIUrl":"https://doi.org/10.1515/anona-2023-0119","url":null,"abstract":"\u0000 In this article, we consider the bounded variation capacity (BV capacity) and characterize the Sobolev-type inequalities related to BV functions in a general framework of strictly local Dirichlet spaces with a doubling measure via the BV capacity. Under a weak Bakry-Émery curvature-type condition, we give the connection between the Hausdorff measure and the Hausdorff capacity, and discover some capacitary inequalities and Maz’ya-Sobolev inequalities for BV functions. The De Giorgi characterization for total variation is also obtained with a quasi-Bakry-Émery curvature condition. It should be noted that the results in this article are proved if the Dirichlet space supports the weak \u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 1\u0000 ,\u0000 2\u0000 \u0000 )\u0000 \u0000 \u0000 left(1,2)\u0000 \u0000 -Poincaré inequality instead of the weak \u0000 \u0000 \u0000 \u0000 \u0000 (\u0000 \u0000 1\u0000 ,\u0000 1\u0000 \u0000 )\u0000 \u0000 \u0000 left(1,1)\u0000 \u0000 -Poincaré inequality compared with the results in the previous references.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140526306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system","authors":"Jian Zhang, Huitao Zhou, Heilong Mi","doi":"10.1515/anona-2023-0139","DOIUrl":"https://doi.org/10.1515/anona-2023-0139","url":null,"abstract":"\u0000 <jats:p>This article is concerned with the following Hamiltonian elliptic system: <jats:disp-formula id=\"j_anona-2023-0139_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0139_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\u0000 <m:mfenced open=\"{\" close=\"\">\u0000 <m:mrow>\u0000 <m:mtable displaystyle=\"true\">\u0000 <m:mtr>\u0000 <m:mtd columnalign=\"left\">\u0000 <m:mo>−</m:mo>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi>ε</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mn>2</m:mn>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mi mathvariant=\"normal\">Δ</m:mi>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>+</m:mo>\u0000 <m:mi>ε</m:mi>\u0000 <m:mover accent=\"true\">\u0000 <m:mrow>\u0000 <m:mi>b</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mo>→</m:mo>\u0000 </m:mrow>\u0000 </m:mover>\u0000 <m:mo>⋅</m:mo>\u0000 <m:mrow>\u0000 <m:mo>∇</m:mo>\u0000 </m:mrow>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>+</m:mo>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>+</m:mo>\u0000 <m:mi>V</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>x</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:mi>v</m:mi>\u0000 <m:mo>=</m:mo>\u0000 <m:msub>\u0000 ","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140527107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities","authors":"Yue Jia, Xianyong Yang","doi":"10.1515/anona-2023-0130","DOIUrl":"https://doi.org/10.1515/anona-2023-0130","url":null,"abstract":"\u0000 <jats:p>In this article, we study the following quasilinear equation with nonlocal nonlinearity <jats:disp-formula id=\"j_anona-2023-0130_eq_001\">\u0000 <jats:alternatives>\u0000 <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_anona-2023-0130_eq_001.png\" />\u0000 <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\">\u0000 <m:mo>−</m:mo>\u0000 <m:mi mathvariant=\"normal\">Δ</m:mi>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>−</m:mo>\u0000 <m:mi>κ</m:mi>\u0000 <m:mi>u</m:mi>\u0000 <m:mi>Δ</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mn>2</m:mn>\u0000 </m:mrow>\u0000 </m:msup>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:mo>+</m:mo>\u0000 <m:mi>λ</m:mi>\u0000 <m:mi>u</m:mi>\u0000 <m:mo>=</m:mo>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:msup>\u0000 <m:mrow>\u0000 <m:mo>∣</m:mo>\u0000 <m:mi>x</m:mi>\u0000 <m:mo>∣</m:mo>\u0000 </m:mrow>\u0000 <m:mrow>\u0000 <m:mo>−</m:mo>\u0000 <m:mi>μ</m:mi>\u0000 </m:mrow>\u0000 </m:msup>\u0000 <m:mo>*</m:mo>\u0000 <m:mi>F</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 </m:mrow>\u0000 <m:mo>)</m:mo>\u0000 </m:mrow>\u0000 <m:mi>f</m:mi>\u0000 <m:mrow>\u0000 <m:mo>(</m:mo>\u0000 <m:mrow>\u0000 <m:mi>u</m:mi>\u0000 </m:mrow>\u0000 ","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140522614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Higher integrability for anisotropic parabolic systems of p-Laplace type","authors":"Leon Mons","doi":"10.1515/anona-2022-0308","DOIUrl":"https://doi.org/10.1515/anona-2022-0308","url":null,"abstract":"Abstract In this article, we consider anisotropic parabolic systems of p p -Laplace type. The model case is the parabolic p i {p}_{i} -Laplace system u t − ∑ i = 1 n ∂ ∂ x i ( ∣ D i u ∣ p i − 2 D i u ) = 0 {u}_{t}-mathop{sum }limits_{i=1}^{n}frac{partial }{partial {x}_{i}}({| {D}_{i}u| }^{{p}_{i}-2}{D}_{i}u)=0 with exponents p i ≥ 2 {p}_{i}ge 2 . Under the assumption that the exponents are not too far apart, i.e., the difference p max − p min {p}_{max }-{p}_{min } is sufficiently small, we establish a higher integrability result for weak solutions. This extends a result, which was only known for the elliptic setting, to the parabolic setting.","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45131011","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dirichlet problems involving the Hardy-Leray operators with multiple polars","authors":"Huyuan Chen, Xiaowei Chen","doi":"10.1515/anona-2022-0320","DOIUrl":"https://doi.org/10.1515/anona-2022-0320","url":null,"abstract":"Abstract Our aim of this article is to study qualitative properties of Dirichlet problems involving the Hardy-Leray operator ℒ V ≔ − Δ + V {{mathcal{ {mathcal L} }}}_{V}:= -Delta +V , where V ( x ) = ∑ i = 1 m μ i ∣ x − A i ∣ 2 Vleft(x)={sum }_{i=1}^{m}frac{{mu }_{i}}{{| x-{A}_{i}| }^{2}} , with μ i ≥ − ( N − 2 ) 2 4 {mu }_{i}ge -frac{{left(N-2)}^{2}}{4} being the Hardy-Leray potential containing the polars’ set A m = { A i : i = 1 , … , m } {{mathcal{A}}}_{m}=left{{A}_{i}:i=1,ldots ,mright} in R N {{mathbb{R}}}^{N} ( N ≥ 2 Nge 2 ). Since the inverse-square potentials are critical with respect to the Laplacian operator, the coefficients { μ i } i = 1 m {left{{mu }_{i}right}}_{i=1}^{m} and the locations of polars { A i } left{{A}_{i}right} play an important role in the properties of solutions to the related Poisson problems subject to zero Dirichlet boundary conditions. Let Ω Omega be a bounded domain containing A m {{mathcal{A}}}_{m} . First, we obtain increasing Dirichlet eigenvalues: ℒ V u = λ u in Ω , u = 0 on ∂ Ω , {{mathcal{ {mathcal L} }}}_{V}u=lambda uhspace{1.0em}{rm{in}}hspace{0.33em}Omega ,hspace{1.0em}u=0hspace{1.0em}{rm{on}}hspace{0.33em}partial Omega , and the positivity of the principle eigenvalue depends on the strength μ i {mu }_{i} and polars’ setting. When the spectral does not contain the origin, we then consider the weak solutions of the Poisson problem ( E ) ℒ V u = ν in Ω , u = 0 on ∂ Ω , left(E)hspace{1.0em}hspace{1.0em}{{mathcal{ {mathcal L} }}}_{V}u=nu hspace{1em}{rm{in}}hspace{0.33em}Omega ,hspace{1.0em}u=0hspace{1em}{rm{on}}hspace{0.33em}partial Omega , when ν nu belongs to L p ( Ω ) {L}^{p}left(Omega ) , with p > 2 N N + 2 pgt frac{2N}{N+2} in the variational framework, and we obtain a global weighted L ∞ {L}^{infty } estimate when p > N 2 pgt frac{N}{2} . When the principle eigenvalue is positive and ν nu is a Radon measure, we build a weighted distributional framework to show the existence of weak solutions of problem ( E ) left(E) . Moreover, via this weighted distributional framework, we can obtain a sharp assumption of ν ∈ C γ ( Ω ¯ A m ) nu in {{mathcal{C}}}^{gamma }left(bar{Omega }setminus {{mathcal{A}}}_{m}) for the existence of isolated singular solutions for problem ( E ) left(E) .","PeriodicalId":51301,"journal":{"name":"Advances in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":4.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43333591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}