{"title":"一类具有临界指数和测量数据的奇异椭圆型问题正解的存在性","authors":"A. Panda, D. Choudhuri, R. K. Giri","doi":"10.1216/rmj.2021.51.973","DOIUrl":null,"url":null,"abstract":"We prove the existence of a positive {\\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \\begin{equation} \\begin{split} (-\\Delta)^su&= \\frac{1}{u^\\gamma}+\\lambda u^{2_s^*-1}+\\mu ~\\text{in}~\\Omega, u&>0~\\text{in}~\\Omega, u&= 0~\\text{in}~\\mathbb{R}^N\\setminus\\Omega. \\end{split} \\end{equation} Here, $\\Omega$ is a bounded domain of $\\mathbb{R}^N$, $s\\in (0,1)$, $2s<N$, $\\lambda,\\gamma\\in (0,1)$, $2_s^*=\\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $\\mu$ is a nonnegative bounded Radon measure in $\\Omega$.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":"33 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Existence of positive solutions for a singular elliptic problem with critical exponent and measure data\",\"authors\":\"A. Panda, D. Choudhuri, R. K. Giri\",\"doi\":\"10.1216/rmj.2021.51.973\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove the existence of a positive {\\\\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \\\\begin{equation} \\\\begin{split} (-\\\\Delta)^su&= \\\\frac{1}{u^\\\\gamma}+\\\\lambda u^{2_s^*-1}+\\\\mu ~\\\\text{in}~\\\\Omega, u&>0~\\\\text{in}~\\\\Omega, u&= 0~\\\\text{in}~\\\\mathbb{R}^N\\\\setminus\\\\Omega. \\\\end{split} \\\\end{equation} Here, $\\\\Omega$ is a bounded domain of $\\\\mathbb{R}^N$, $s\\\\in (0,1)$, $2s<N$, $\\\\lambda,\\\\gamma\\\\in (0,1)$, $2_s^*=\\\\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $\\\\mu$ is a nonnegative bounded Radon measure in $\\\\Omega$.\",\"PeriodicalId\":8445,\"journal\":{\"name\":\"arXiv: Analysis of PDEs\",\"volume\":\"33 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv: Analysis of PDEs\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1216/rmj.2021.51.973\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Analysis of PDEs","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1216/rmj.2021.51.973","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Existence of positive solutions for a singular elliptic problem with critical exponent and measure data
We prove the existence of a positive {\it SOLA (Solutions Obtained as Limits of Approximations)} to the following PDE involving fractional power of Laplacian \begin{equation} \begin{split} (-\Delta)^su&= \frac{1}{u^\gamma}+\lambda u^{2_s^*-1}+\mu ~\text{in}~\Omega, u&>0~\text{in}~\Omega, u&= 0~\text{in}~\mathbb{R}^N\setminus\Omega. \end{split} \end{equation} Here, $\Omega$ is a bounded domain of $\mathbb{R}^N$, $s\in (0,1)$, $2s
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