涉及幂非线性和具有边界奇点的哈代势的半线性椭圆方程

IF 1.3 3区 数学 Q1 MATHEMATICS
Konstantinos T. Gkikas, Phuoc-Tai Nguyen
{"title":"涉及幂非线性和具有边界奇点的哈代势的半线性椭圆方程","authors":"Konstantinos T. Gkikas, Phuoc-Tai Nguyen","doi":"10.1017/prm.2023.122","DOIUrl":null,"url":null,"abstract":"<p>Let <span><span><span data-mathjax-type=\"texmath\"><span>$\\Omega \\subset \\mathbb {R}^N$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline1.png\"/></span></span> (<span><span><span data-mathjax-type=\"texmath\"><span>$N\\geq 3$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline2.png\"/></span></span>) be a <span><span><span data-mathjax-type=\"texmath\"><span>$C^2$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline3.png\"/></span></span> bounded domain and <span><span><span data-mathjax-type=\"texmath\"><span>$\\Sigma \\subset \\partial \\Omega$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline4.png\"/></span></span> be a <span><span><span data-mathjax-type=\"texmath\"><span>$C^2$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline5.png\"/></span></span> compact submanifold without boundary, of dimension <span><span><span data-mathjax-type=\"texmath\"><span>$k$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline6.png\"/></span></span>, <span><span><span data-mathjax-type=\"texmath\"><span>$0\\leq k \\leq N-1$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline7.png\"/></span></span>. We assume that <span><span><span data-mathjax-type=\"texmath\"><span>$\\Sigma = \\{0\\}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline8.png\"/></span></span> if <span><span><span data-mathjax-type=\"texmath\"><span>$k = 0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline9.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$\\Sigma =\\partial \\Omega$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline10.png\"/></span></span> if <span><span><span data-mathjax-type=\"texmath\"><span>$k=N-1$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline11.png\"/></span></span>. Let <span><span><span data-mathjax-type=\"texmath\"><span>$d_{\\Sigma }(x)=\\mathrm {dist}\\,(x,\\Sigma )$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline12.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$L_\\mu = \\Delta + \\mu \\,d_{\\Sigma }^{-2}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline13.png\"/></span></span>, where <span><span><span data-mathjax-type=\"texmath\"><span>$\\mu \\in {\\mathbb {R}}$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline14.png\"/></span></span>. We study boundary value problems (<span><span><span data-mathjax-type=\"texmath\"><span>$P_\\pm$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline15.png\"/></span></span>) <span><span><span data-mathjax-type=\"texmath\"><span>$-{L_\\mu} u \\pm |u|^{p-1}u = 0$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline16.png\"/></span></span> in <span><span><span data-mathjax-type=\"texmath\"><span>$\\Omega$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline17.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$\\mathrm {tr}_{\\mu,\\Sigma}(u)=\\nu$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline18.png\"/></span></span> on <span><span><span data-mathjax-type=\"texmath\"><span>$\\partial \\Omega$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline19.png\"/></span></span>, where <span><span><span data-mathjax-type=\"texmath\"><span>$p&gt;1$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline20.png\"/></span></span>, <span><span><span data-mathjax-type=\"texmath\"><span>$\\nu$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline21.png\"/></span></span> is a given measure on <span><span><span data-mathjax-type=\"texmath\"><span>$\\partial \\Omega$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline22.png\"/></span></span> and <span><span><span data-mathjax-type=\"texmath\"><span>$\\mathrm {tr}_{\\mu,\\Sigma}(u)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline23.png\"/></span></span> denotes the boundary trace of <span><span><span data-mathjax-type=\"texmath\"><span>$u$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline24.png\"/></span></span> associated to <span><span><span data-mathjax-type=\"texmath\"><span>$L_\\mu$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline25.png\"/></span></span>. Different critical exponents for the existence of a solution to (<span><span><span data-mathjax-type=\"texmath\"><span>$P_\\pm$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline26.png\"/></span></span>) appear according to concentration of <span><span><span data-mathjax-type=\"texmath\"><span>$\\nu$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline27.png\"/></span></span>. The solvability for problem (<span><span><span data-mathjax-type=\"texmath\"><span>$P_+$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline28.png\"/></span></span>) was proved in [3, 29] in subcritical ranges for <span><span><span data-mathjax-type=\"texmath\"><span>$p$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline29.png\"/></span></span>, namely for <span><span><span data-mathjax-type=\"texmath\"><span>$p$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline30.png\"/></span></span> smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of <span><span><span data-mathjax-type=\"texmath\"><span>$-L_\\mu$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline31.png\"/></span></span>, we provide conditions on <span><span><span data-mathjax-type=\"texmath\"><span>$\\nu$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline32.png\"/></span></span> expressed in terms of capacities for the existence of a (unique) solution to (<span><span><span data-mathjax-type=\"texmath\"><span>$P_+$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline33.png\"/></span></span>) in supercritical ranges for <span><span><span data-mathjax-type=\"texmath\"><span>$p$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline34.png\"/></span></span>, i.e. for <span><span><span data-mathjax-type=\"texmath\"><span>$p$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline35.png\"/></span></span> equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to (<span><span><span data-mathjax-type=\"texmath\"><span>$P_-$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline36.png\"/></span></span>) under a smallness assumption on <span><span><span data-mathjax-type=\"texmath\"><span>$\\nu$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline37.png\"/></span></span>.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"35 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities\",\"authors\":\"Konstantinos T. Gkikas, Phuoc-Tai Nguyen\",\"doi\":\"10.1017/prm.2023.122\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\Omega \\\\subset \\\\mathbb {R}^N$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline1.png\\\"/></span></span> (<span><span><span data-mathjax-type=\\\"texmath\\\"><span>$N\\\\geq 3$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline2.png\\\"/></span></span>) be a <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$C^2$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline3.png\\\"/></span></span> bounded domain and <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\Sigma \\\\subset \\\\partial \\\\Omega$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline4.png\\\"/></span></span> be a <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$C^2$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline5.png\\\"/></span></span> compact submanifold without boundary, of dimension <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$k$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline6.png\\\"/></span></span>, <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$0\\\\leq k \\\\leq N-1$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline7.png\\\"/></span></span>. We assume that <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\Sigma = \\\\{0\\\\}$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline8.png\\\"/></span></span> if <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$k = 0$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline9.png\\\"/></span></span> and <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\Sigma =\\\\partial \\\\Omega$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline10.png\\\"/></span></span> if <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$k=N-1$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline11.png\\\"/></span></span>. Let <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$d_{\\\\Sigma }(x)=\\\\mathrm {dist}\\\\,(x,\\\\Sigma )$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline12.png\\\"/></span></span> and <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$L_\\\\mu = \\\\Delta + \\\\mu \\\\,d_{\\\\Sigma }^{-2}$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline13.png\\\"/></span></span>, where <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mu \\\\in {\\\\mathbb {R}}$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline14.png\\\"/></span></span>. We study boundary value problems (<span><span><span data-mathjax-type=\\\"texmath\\\"><span>$P_\\\\pm$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline15.png\\\"/></span></span>) <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$-{L_\\\\mu} u \\\\pm |u|^{p-1}u = 0$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline16.png\\\"/></span></span> in <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\Omega$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline17.png\\\"/></span></span> and <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathrm {tr}_{\\\\mu,\\\\Sigma}(u)=\\\\nu$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline18.png\\\"/></span></span> on <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\partial \\\\Omega$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline19.png\\\"/></span></span>, where <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$p&gt;1$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline20.png\\\"/></span></span>, <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\nu$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline21.png\\\"/></span></span> is a given measure on <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\partial \\\\Omega$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline22.png\\\"/></span></span> and <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathrm {tr}_{\\\\mu,\\\\Sigma}(u)$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline23.png\\\"/></span></span> denotes the boundary trace of <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$u$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline24.png\\\"/></span></span> associated to <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$L_\\\\mu$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline25.png\\\"/></span></span>. Different critical exponents for the existence of a solution to (<span><span><span data-mathjax-type=\\\"texmath\\\"><span>$P_\\\\pm$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline26.png\\\"/></span></span>) appear according to concentration of <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\nu$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline27.png\\\"/></span></span>. The solvability for problem (<span><span><span data-mathjax-type=\\\"texmath\\\"><span>$P_+$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline28.png\\\"/></span></span>) was proved in [3, 29] in subcritical ranges for <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$p$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline29.png\\\"/></span></span>, namely for <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$p$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline30.png\\\"/></span></span> smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$-L_\\\\mu$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline31.png\\\"/></span></span>, we provide conditions on <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\nu$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline32.png\\\"/></span></span> expressed in terms of capacities for the existence of a (unique) solution to (<span><span><span data-mathjax-type=\\\"texmath\\\"><span>$P_+$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline33.png\\\"/></span></span>) in supercritical ranges for <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$p$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline34.png\\\"/></span></span>, i.e. for <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$p$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline35.png\\\"/></span></span> equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to (<span><span><span data-mathjax-type=\\\"texmath\\\"><span>$P_-$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline36.png\\\"/></span></span>) under a smallness assumption on <span><span><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\nu$</span></span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231220091413004-0011:S0308210523001221:S0308210523001221_inline37.png\\\"/></span></span>.</p>\",\"PeriodicalId\":54560,\"journal\":{\"name\":\"Proceedings of the Royal Society of Edinburgh Section A-Mathematics\",\"volume\":\"35 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2023-12-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Royal Society of Edinburgh Section A-Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/prm.2023.122\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/prm.2023.122","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

让 $\Omega \subset \mathbb {R}^N$ ($N\geq 3$) 是一个 $C^2$ 有界域,并且 $\Sigma \subset \partial \Omega$ 是一个无边界的 $C^2$ 紧凑子漫游,维数为 $k$, $0\leq k \leq N-1$。我们假设,如果 $k = 0$,$\Sigma = \{0\}$;如果 $k=N-1$,$\Sigma = \partial \Omega$。让 $d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ 和 $L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$, 其中 $\mu \ in {\mathbb {R}}$.我们研究了 $-{L_\mu} u \pm |u|^{p-1}u = 0$ in $\Omega$ 和 $\mathrm {tr}_{\mu,\Sigma}(u)=\nu$ on $partial \Omega$ 的边界值问题($P_\pm$),其中 $p>;1$,$\nu$是$partial \Omega$上的给定度量,$mathrm {tr}_{mu,\Sigma}(u)$ 表示与$L_\mu$相关的$u$的边界迹。根据 $\nu$ 的浓度不同,($P_\pm$)解存在的临界指数也不同。文献[3, 29]证明了问题($P_+$)在 $p$ 的亚临界范围内的可解性,即在 $p$ 小于临界指数之一的情况下。在本文中,假定 $-L_\mu$ 的第一个特征值为正值,我们提供了在 $p$ 的超临界范围内,即在 $p$ 等于或大于临界指数之一时,以容量表示的 $\nu$ 上存在 ($P_+$) (唯一)解的条件。我们还建立了在 $\nu$ 小性假设下 ($P_-$) 解存在的各种等价标准。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Semilinear elliptic equations involving power nonlinearities and Hardy potentials with boundary singularities

Let $\Omega \subset \mathbb {R}^N$ ($N\geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \partial \Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0\leq k \leq N-1$. We assume that $\Sigma = \{0\}$ if $k = 0$ and $\Sigma =\partial \Omega$ if $k=N-1$. Let $d_{\Sigma }(x)=\mathrm {dist}\,(x,\Sigma )$ and $L_\mu = \Delta + \mu \,d_{\Sigma }^{-2}$, where $\mu \in {\mathbb {R}}$. We study boundary value problems ($P_\pm$) $-{L_\mu} u \pm |u|^{p-1}u = 0$ in $\Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)=\nu$ on $\partial \Omega$, where $p>1$, $\nu$ is a given measure on $\partial \Omega$ and $\mathrm {tr}_{\mu,\Sigma}(u)$ denotes the boundary trace of $u$ associated to $L_\mu$. Different critical exponents for the existence of a solution to ($P_\pm$) appear according to concentration of $\nu$. The solvability for problem ($P_+$) was proved in [3, 29] in subcritical ranges for $p$, namely for $p$ smaller than one of the critical exponents. In this paper, assuming the positivity of the first eigenvalue of $-L_\mu$, we provide conditions on $\nu$ expressed in terms of capacities for the existence of a (unique) solution to ($P_+$) in supercritical ranges for $p$, i.e. for $p$ equal or bigger than one of the critical exponents. We also establish various equivalent criteria for the existence of a solution to ($P_-$) under a smallness assumption on $\nu$.

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来源期刊
CiteScore
3.00
自引率
0.00%
发文量
72
审稿时长
6-12 weeks
期刊介绍: A flagship publication of The Royal Society of Edinburgh, Proceedings A is a prestigious, general mathematics journal publishing peer-reviewed papers of international standard across the whole spectrum of mathematics, but with the emphasis on applied analysis and differential equations. An international journal, publishing six issues per year, Proceedings A has been publishing the highest-quality mathematical research since 1884. Recent issues have included a wealth of key contributors and considered research papers.
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