{"title":"一个具有谱Neumann分数拉普拉斯算子的临界双线性问题的可解性","authors":"N. Ustinov","doi":"10.1090/spmj/1693","DOIUrl":null,"url":null,"abstract":"<p>Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega element-of upper C squared colon\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:msup>\n <mml:mi>C</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:mo>:</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\Omega \\in C^2:</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis negative normal upper Delta right-parenthesis Subscript upper S p Superscript s Baseline u left-parenthesis x right-parenthesis plus u left-parenthesis x right-parenthesis equals u Superscript 2 Super Subscript s Super Superscript asterisk Superscript minus 1 Baseline left-parenthesis x right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\n <mml:msubsup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>S</mml:mi>\n <mml:mi>p</mml:mi>\n </mml:mrow>\n <mml:mi>s</mml:mi>\n </mml:msubsup>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>+</mml:mo>\n <mml:mi>u</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:msup>\n <mml:mi>u</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:msubsup>\n <mml:mn>2</mml:mn>\n <mml:mi>s</mml:mi>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:msubsup>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>x</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(-\\Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Here <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis negative normal upper Delta right-parenthesis Subscript upper S p Superscript s\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi>\n <mml:msubsup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>S</mml:mi>\n <mml:mi>p</mml:mi>\n </mml:mrow>\n <mml:mi>s</mml:mi>\n </mml:msubsup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(-\\Delta )_{Sp}^s</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> stands for the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"s\">\n <mml:semantics>\n <mml:mi>s</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">s</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>th power of the conventional Neumann Laplacian in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Omega double-subset double-struck upper R Superscript n\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi mathvariant=\"normal\">Ω<!-- Ω --></mml:mi>\n <mml:mo>⋐<!-- ⋐ --></mml:mo>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mi>n</mml:mi>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\Omega \\Subset \\mathbb {R}^n</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than-or-equal-to 3\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>n</mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>3</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">n \\geq 3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"s element-of left-parenthesis 0 comma 1 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>s</mml:mi>\n <mml:mo>∈<!-- ∈ --></mml:mo>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>0</mml:mn>\n <mml:mo>,</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">s \\in (0, 1)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 Subscript s Superscript asterisk Baseline equals 2 n slash left-parenthesis n minus 2 s right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msubsup>\n <mml:mn>2</mml:mn>\n <mml:mi>s</mml:mi>\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:msubsup>\n <mml:mo>=</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mi>n</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>/</mml:mo>\n </mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>n</mml:mi>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mi>s</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">2^*_s = 2n/(n-2s)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. For the local case where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"s equals 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>s</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">s = 1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, corresponding results were obtained earlier for the Neumann Laplacian and Neumann <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\n <mml:semantics>\n <mml:mi>p</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-Laplacian operators.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Solvability of a critical semilinear problem with the spectral Neumann fractional Laplacian\",\"authors\":\"N. Ustinov\",\"doi\":\"10.1090/spmj/1693\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Omega element-of upper C squared colon\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi mathvariant=\\\"normal\\\">Ω<!-- Ω --></mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:msup>\\n <mml:mi>C</mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n <mml:mo>:</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Omega \\\\in C^2:</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis negative normal upper Delta right-parenthesis Subscript upper S p Superscript s Baseline u left-parenthesis x right-parenthesis plus u left-parenthesis x right-parenthesis equals u Superscript 2 Super Subscript s Super Superscript asterisk Superscript minus 1 Baseline left-parenthesis x right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mi mathvariant=\\\"normal\\\">Δ<!-- Δ --></mml:mi>\\n <mml:msubsup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>S</mml:mi>\\n <mml:mi>p</mml:mi>\\n </mml:mrow>\\n <mml:mi>s</mml:mi>\\n </mml:msubsup>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>+</mml:mo>\\n <mml:mi>u</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mo>=</mml:mo>\\n <mml:msup>\\n <mml:mi>u</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:msubsup>\\n <mml:mn>2</mml:mn>\\n <mml:mi>s</mml:mi>\\n <mml:mo>∗<!-- ∗ --></mml:mo>\\n </mml:msubsup>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>x</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">(-\\\\Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Here <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis negative normal upper Delta right-parenthesis Subscript upper S p Superscript s\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mi mathvariant=\\\"normal\\\">Δ<!-- Δ --></mml:mi>\\n <mml:msubsup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>S</mml:mi>\\n <mml:mi>p</mml:mi>\\n </mml:mrow>\\n <mml:mi>s</mml:mi>\\n </mml:msubsup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">(-\\\\Delta )_{Sp}^s</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> stands for the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"s\\\">\\n <mml:semantics>\\n <mml:mi>s</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">s</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>th power of the conventional Neumann Laplacian in <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper Omega double-subset double-struck upper R Superscript n\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi mathvariant=\\\"normal\\\">Ω<!-- Ω --></mml:mi>\\n <mml:mo>⋐<!-- ⋐ --></mml:mo>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mi>n</mml:mi>\\n </mml:msup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\Omega \\\\Subset \\\\mathbb {R}^n</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n greater-than-or-equal-to 3\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>n</mml:mi>\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>3</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">n \\\\geq 3</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"s element-of left-parenthesis 0 comma 1 right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>s</mml:mi>\\n <mml:mo>∈<!-- ∈ --></mml:mo>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mn>0</mml:mn>\\n <mml:mo>,</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">s \\\\in (0, 1)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"2 Subscript s Superscript asterisk Baseline equals 2 n slash left-parenthesis n minus 2 s right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:msubsup>\\n <mml:mn>2</mml:mn>\\n <mml:mi>s</mml:mi>\\n <mml:mo>∗<!-- ∗ --></mml:mo>\\n </mml:msubsup>\\n <mml:mo>=</mml:mo>\\n <mml:mn>2</mml:mn>\\n <mml:mi>n</mml:mi>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>/</mml:mo>\\n </mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mi>n</mml:mi>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mn>2</mml:mn>\\n <mml:mi>s</mml:mi>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">2^*_s = 2n/(n-2s)</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. For the local case where <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"s equals 1\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>s</mml:mi>\\n <mml:mo>=</mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">s = 1</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, corresponding results were obtained earlier for the Neumann Laplacian and Neumann <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p\\\">\\n <mml:semantics>\\n <mml:mi>p</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-Laplacian operators.</p>\",\"PeriodicalId\":51162,\"journal\":{\"name\":\"St Petersburg Mathematical Journal\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2021-12-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"St Petersburg Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/spmj/1693\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1693","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 3
摘要
给出了Ω∈C2:\Omega\中的分数阶Sobolev不等式在C^2中产生的问题基态解存在的充分条件:(−Δ)S p S u(x)+u(x=u 2 s*−1(x)(-\Δ)_{Sp}^s u(x)+u(x)=u ^{2^*_s-1}(x)。这里(−Δ)S p S(-\Delta)_{Sp}^S代表传统Neumann拉普拉斯算子在Ω中的S次幂,s∈(0,1)s\在(0,l)中,2s*=2 n/(n−2s)2^*_s=2n/(n-2s)。对于s=1 s=1的局部情况,Neumann-拉普拉斯算子和Neumann-p-拉普拉斯算子的相应结果早些时候得到。
Solvability of a critical semilinear problem with the spectral Neumann fractional Laplacian
Sufficient conditions are provided for the existence of a ground state solution for the problem generated by the fractional Sobolev inequality in Ω∈C2:\Omega \in C^2:(−Δ)Spsu(x)+u(x)=u2s∗−1(x)(-\Delta )_{Sp}^s u(x) + u(x) = u^{2^*_s-1}(x). Here (−Δ)Sps(-\Delta )_{Sp}^s stands for the ssth power of the conventional Neumann Laplacian in Ω⋐Rn\Omega \Subset \mathbb {R}^n, n≥3n \geq 3, s∈(0,1)s \in (0, 1), 2s∗=2n/(n−2s)2^*_s = 2n/(n-2s). For the local case where s=1s = 1, corresponding results were obtained earlier for the Neumann Laplacian and Neumann pp-Laplacian operators.
期刊介绍:
This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.