Existence and Multiplicity of Solutions for Nonhomogeneous Schrödinger–Kirchhoff-Type Fourth-Order Elliptic Equations in RN View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2019-08-19

AUTHORS

Jiabin Zuo, Tianqing An, Yuanfang Ru, Dafang Zhao

ABSTRACT

In this article, we study the following nonhomogeneous Schrödinger–Kirchhoff-type equation Δ2u-(a+b∫RN|∇u|2dx)Δu+V(x)u=f(x,u)+h(x)inRN,u∈H2(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{cl} &{}\displaystyle \Delta ^{2}u-(a+b\int _{\mathbb {R}^{N}}|\nabla u|^{2}\mathrm{d}x)\Delta u+V(x)u=f(x,u)+h(x)\;\; \text {in }\;\; \mathbb {R}^{N}, \\ &{}\displaystyle u \in H^{2}(\mathbb {R}^{N}), \end{array}\right. \end{aligned}$$\end{document}where a>0,b≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0,b\ge 0$$\end{document}. Under the suitable assumptions of V(x), f(x, u), and h(x), we prove the existence of nontrivial solution using the Mountain Pass Theorem. In addition, infinitely many high-energy solutions are obtained by two kinds of methods (i.e., Symmetry Mountain Pass Theorem and Fountain Theorem) when h(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x)=0$$\end{document}. Moreover, we also show infinitely many radial solutions of this equation. More... »

PAGES

123

References to SciGraph publications

  • 1996. Minimax Theorems in NONE
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00009-019-1402-2

    DOI

    http://dx.doi.org/10.1007/s00009-019-1402-2

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1120409935


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "School of Applied Sciences, Jilin Engineering Normal University, 130052, Changchun, People\u2019s Republic of China", 
              "id": "http://www.grid.ac/institutes/grid.443318.9", 
              "name": [
                "College of Science, Hohai University, 210098, Nanjing, People\u2019s Republic of China", 
                "School of Applied Sciences, Jilin Engineering Normal University, 130052, Changchun, People\u2019s Republic of China"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Zuo", 
            "givenName": "Jiabin", 
            "id": "sg:person.016713122037.01", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016713122037.01"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "College of Science, Hohai University, 210098, Nanjing, People\u2019s Republic of China", 
              "id": "http://www.grid.ac/institutes/grid.257065.3", 
              "name": [
                "College of Science, Hohai University, 210098, Nanjing, People\u2019s Republic of China"
              ], 
              "type": "Organization"
            }, 
            "familyName": "An", 
            "givenName": "Tianqing", 
            "id": "sg:person.012416633477.36", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012416633477.36"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "College of Science, China Pharmaceutical University, 211198, Nanjing, People\u2019s Republic of China", 
              "id": "http://www.grid.ac/institutes/None", 
              "name": [
                "College of Science, China Pharmaceutical University, 211198, Nanjing, People\u2019s Republic of China"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Ru", 
            "givenName": "Yuanfang", 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "School of Mathematics and Statistics, Hubei Normal University, 435002, Huangshi, People\u2019s Republic of China", 
              "id": "http://www.grid.ac/institutes/grid.462271.4", 
              "name": [
                "College of Science, Hohai University, 210098, Nanjing, People\u2019s Republic of China", 
                "School of Mathematics and Statistics, Hubei Normal University, 435002, Huangshi, People\u2019s Republic of China"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Zhao", 
            "givenName": "Dafang", 
            "id": "sg:person.012305220221.44", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012305220221.44"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/978-1-4612-4146-1", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1022709270", 
              "https://doi.org/10.1007/978-1-4612-4146-1"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2019-08-19", 
        "datePublishedReg": "2019-08-19", 
        "description": "In this article, we study the following nonhomogeneous Schr\u00f6dinger\u2013Kirchhoff-type equation \u03942u-(a+b\u222bRN|\u2207u|2dx)\u0394u+V(x)u=f(x,u)+h(x)inRN,u\u2208H2(RN),\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{cl} &{}\\displaystyle \\Delta ^{2}u-(a+b\\int _{\\mathbb {R}^{N}}|\\nabla u|^{2}\\mathrm{d}x)\\Delta u+V(x)u=f(x,u)+h(x)\\;\\; \\text {in }\\;\\; \\mathbb {R}^{N}, \\\\ &{}\\displaystyle u \\in H^{2}(\\mathbb {R}^{N}), \\end{array}\\right. \\end{aligned}$$\\end{document}where a>0,b\u22650\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$a>0,b\\ge 0$$\\end{document}. Under the suitable assumptions of V(x), f(x,\u00a0u), and h(x), we prove the existence of nontrivial solution using the Mountain Pass Theorem. In addition, infinitely many high-energy solutions are obtained by two kinds of methods (i.e., Symmetry Mountain Pass Theorem and Fountain Theorem) when h(x)=0\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$h(x)=0$$\\end{document}. Moreover, we also show infinitely many radial solutions of this equation.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s00009-019-1402-2", 
        "isAccessibleForFree": false, 
        "isPartOf": [
          {
            "id": "sg:journal.1135886", 
            "issn": [
              "1660-5446", 
              "1660-5454"
            ], 
            "name": "Mediterranean Journal of Mathematics", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "5", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "16"
          }
        ], 
        "keywords": [
          "fourth-order elliptic equations", 
          "Schr\u00f6dinger\u2013Kirchhoff", 
          "multiplicity of solutions", 
          "mountain pass theorem", 
          "elliptic equations", 
          "nontrivial solutions", 
          "suitable assumptions", 
          "high energy solutions", 
          "radial solutions", 
          "equations", 
          "kinds of methods", 
          "solution", 
          "theorem", 
          "existence", 
          "\u03942u", 
          "Rn", 
          "assumption", 
          "multiplicity", 
          "kind", 
          "article", 
          "addition", 
          "method"
        ], 
        "name": "Existence and Multiplicity of Solutions for Nonhomogeneous Schr\u00f6dinger\u2013Kirchhoff-Type Fourth-Order Elliptic Equations in RN", 
        "pagination": "123", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1120409935"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s00009-019-1402-2"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s00009-019-1402-2", 
          "https://app.dimensions.ai/details/publication/pub.1120409935"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-11-24T21:05", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20221124/entities/gbq_results/article/article_827.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s00009-019-1402-2"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00009-019-1402-2'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00009-019-1402-2'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00009-019-1402-2'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00009-019-1402-2'


     

    This table displays all metadata directly associated to this object as RDF triples.

    114 TRIPLES      21 PREDICATES      46 URIs      37 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00009-019-1402-2 schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author N048ea2fd69094c5c895e10c5d8a3d27d
    4 schema:citation sg:pub.10.1007/978-1-4612-4146-1
    5 schema:datePublished 2019-08-19
    6 schema:datePublishedReg 2019-08-19
    7 schema:description In this article, we study the following nonhomogeneous Schrödinger–Kirchhoff-type equation Δ2u-(a+b∫RN|∇u|2dx)Δu+V(x)u=f(x,u)+h(x)inRN,u∈H2(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{cl} &{}\displaystyle \Delta ^{2}u-(a+b\int _{\mathbb {R}^{N}}|\nabla u|^{2}\mathrm{d}x)\Delta u+V(x)u=f(x,u)+h(x)\;\; \text {in }\;\; \mathbb {R}^{N}, \\ &{}\displaystyle u \in H^{2}(\mathbb {R}^{N}), \end{array}\right. \end{aligned}$$\end{document}where a>0,b≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>0,b\ge 0$$\end{document}. Under the suitable assumptions of V(x), f(x, u), and h(x), we prove the existence of nontrivial solution using the Mountain Pass Theorem. In addition, infinitely many high-energy solutions are obtained by two kinds of methods (i.e., Symmetry Mountain Pass Theorem and Fountain Theorem) when h(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h(x)=0$$\end{document}. Moreover, we also show infinitely many radial solutions of this equation.
    8 schema:genre article
    9 schema:isAccessibleForFree false
    10 schema:isPartOf N647fa06aa8d546d5b08d05298f63bc84
    11 Ncb52be84db0c4d16be5ae107ac2df980
    12 sg:journal.1135886
    13 schema:keywords Rn
    14 Schrödinger–Kirchhoff
    15 addition
    16 article
    17 assumption
    18 elliptic equations
    19 equations
    20 existence
    21 fourth-order elliptic equations
    22 high energy solutions
    23 kind
    24 kinds of methods
    25 method
    26 mountain pass theorem
    27 multiplicity
    28 multiplicity of solutions
    29 nontrivial solutions
    30 radial solutions
    31 solution
    32 suitable assumptions
    33 theorem
    34 Δ2u
    35 schema:name Existence and Multiplicity of Solutions for Nonhomogeneous Schrödinger–Kirchhoff-Type Fourth-Order Elliptic Equations in RN
    36 schema:pagination 123
    37 schema:productId N90da8471f760414aab01ad56ec7200dc
    38 Nf8fea94dfb72414f95a58f3c7fcf2873
    39 schema:sameAs https://app.dimensions.ai/details/publication/pub.1120409935
    40 https://doi.org/10.1007/s00009-019-1402-2
    41 schema:sdDatePublished 2022-11-24T21:05
    42 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    43 schema:sdPublisher Nb57b2198c8414b6fae8f5af331e1627b
    44 schema:url https://doi.org/10.1007/s00009-019-1402-2
    45 sgo:license sg:explorer/license/
    46 sgo:sdDataset articles
    47 rdf:type schema:ScholarlyArticle
    48 N048ea2fd69094c5c895e10c5d8a3d27d rdf:first sg:person.016713122037.01
    49 rdf:rest N467fad09cebf4b8d89f5b6f2331cc820
    50 N467fad09cebf4b8d89f5b6f2331cc820 rdf:first sg:person.012416633477.36
    51 rdf:rest Nf1e0d8d0bd1d4a338ceb6c04cb2ea99e
    52 N647fa06aa8d546d5b08d05298f63bc84 schema:issueNumber 5
    53 rdf:type schema:PublicationIssue
    54 N90da8471f760414aab01ad56ec7200dc schema:name doi
    55 schema:value 10.1007/s00009-019-1402-2
    56 rdf:type schema:PropertyValue
    57 Nabe29fb332c1460eb064c6b75728be1e schema:affiliation grid-institutes:None
    58 schema:familyName Ru
    59 schema:givenName Yuanfang
    60 rdf:type schema:Person
    61 Nb57b2198c8414b6fae8f5af331e1627b schema:name Springer Nature - SN SciGraph project
    62 rdf:type schema:Organization
    63 Ncb52be84db0c4d16be5ae107ac2df980 schema:volumeNumber 16
    64 rdf:type schema:PublicationVolume
    65 Ncb58db0ca0d64cf9be88c5539ce64134 rdf:first sg:person.012305220221.44
    66 rdf:rest rdf:nil
    67 Nf1e0d8d0bd1d4a338ceb6c04cb2ea99e rdf:first Nabe29fb332c1460eb064c6b75728be1e
    68 rdf:rest Ncb58db0ca0d64cf9be88c5539ce64134
    69 Nf8fea94dfb72414f95a58f3c7fcf2873 schema:name dimensions_id
    70 schema:value pub.1120409935
    71 rdf:type schema:PropertyValue
    72 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    73 schema:name Mathematical Sciences
    74 rdf:type schema:DefinedTerm
    75 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    76 schema:name Pure Mathematics
    77 rdf:type schema:DefinedTerm
    78 sg:journal.1135886 schema:issn 1660-5446
    79 1660-5454
    80 schema:name Mediterranean Journal of Mathematics
    81 schema:publisher Springer Nature
    82 rdf:type schema:Periodical
    83 sg:person.012305220221.44 schema:affiliation grid-institutes:grid.462271.4
    84 schema:familyName Zhao
    85 schema:givenName Dafang
    86 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012305220221.44
    87 rdf:type schema:Person
    88 sg:person.012416633477.36 schema:affiliation grid-institutes:grid.257065.3
    89 schema:familyName An
    90 schema:givenName Tianqing
    91 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012416633477.36
    92 rdf:type schema:Person
    93 sg:person.016713122037.01 schema:affiliation grid-institutes:grid.443318.9
    94 schema:familyName Zuo
    95 schema:givenName Jiabin
    96 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016713122037.01
    97 rdf:type schema:Person
    98 sg:pub.10.1007/978-1-4612-4146-1 schema:sameAs https://app.dimensions.ai/details/publication/pub.1022709270
    99 https://doi.org/10.1007/978-1-4612-4146-1
    100 rdf:type schema:CreativeWork
    101 grid-institutes:None schema:alternateName College of Science, China Pharmaceutical University, 211198, Nanjing, People’s Republic of China
    102 schema:name College of Science, China Pharmaceutical University, 211198, Nanjing, People’s Republic of China
    103 rdf:type schema:Organization
    104 grid-institutes:grid.257065.3 schema:alternateName College of Science, Hohai University, 210098, Nanjing, People’s Republic of China
    105 schema:name College of Science, Hohai University, 210098, Nanjing, People’s Republic of China
    106 rdf:type schema:Organization
    107 grid-institutes:grid.443318.9 schema:alternateName School of Applied Sciences, Jilin Engineering Normal University, 130052, Changchun, People’s Republic of China
    108 schema:name College of Science, Hohai University, 210098, Nanjing, People’s Republic of China
    109 School of Applied Sciences, Jilin Engineering Normal University, 130052, Changchun, People’s Republic of China
    110 rdf:type schema:Organization
    111 grid-institutes:grid.462271.4 schema:alternateName School of Mathematics and Statistics, Hubei Normal University, 435002, Huangshi, People’s Republic of China
    112 schema:name College of Science, Hohai University, 210098, Nanjing, People’s Republic of China
    113 School of Mathematics and Statistics, Hubei Normal University, 435002, Huangshi, People’s Republic of China
    114 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...