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Positive solution sets for nonlinear elliptic boundary value problems of concave-convex mixed type with some singularity

Research Project

Project/Area Number 18K03353
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionIbaraki University

Principal Investigator

Umezu Kenichiro  茨城大学, 基礎自然科学野, 教授 (00295453)

Project Period (FY) 2018-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000)
Fiscal Year 2021: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Keywords非線形楕円型境界値問題 / sublinear非線形性 / concave-convex非線形性 / 非自明非負解 / 変分解析 / 分岐解析 / 優解劣解の構成 / ロバン型境界条件 / 正値解集合 / 正値解の一意性 / 正値解の多重性 / 劣線形性 / 特異性を伴う劣線形項 / concave-convex / アプリオリ評価 / 大域的解構造 / 劣線形 / 正値解 / ロジスティック境界条件 / 一意性と多重性 / ラプラス方程式 / 沿岸漁業収穫モデル / 非線型楕円型方程式 / 劣線形(sublinear) / ロバン境界条件 / p-Laplacian / 正値解の厳密な個数 / 多重性と一意性 / 正値性 / 非線形楕円型偏微分方程式 / 線形化第2固有値 / 非線型関数解析 / 非線形楕円型方程式 / concave非線形性 / 符号不定重み関数 / 強正値解 / sub and supersolutions / 変分法 / 特異的混合型非線形性 / 符号不定係数
Outline of Final Research Achievements

We consider nonlinear elliptic boundary value problems of sublinear nonlinearity with some singularity and evaluate the existence and behavior of the nontrivial nonnegative solutions under the Dirichlet boundary condition as a parameter varies. Then, we consider a class of concave-convex mixed nonlinear problems and prove the existence, nonexistence, and multiplicity of the nontrivial nonnegative solutions. Further, we construct a subcontinuum, i.e., a nonempty, closed, and connected subset, of loop type in the nonnegative solution set. Then, we study the logistic elliptic problem under a sublinear boundary condition with some singularity and clarify the existence, uniqueness, multiplicity, and asymptotic behavior of the nontrivial nonnegative solutions as a parameter varies.

Academic Significance and Societal Importance of the Research Achievements

(1)滑らかな境界をもつ多次元有界領域において,位相的手法を用いてloopをなす解集合の存在を導けた.加えて,線形化固有値問題の第2固有値の解析により,正値解の多重性を伴う可能な限り最小の解曲線を獲得できた.(2)べき乗型非線形項に現れるべき値をパラメータとした非自明非負解の解析は従来の研究ではあまり見られない.べき値がsublinear非線形項の特異性の強さを表現しており,それに応じて非自明非負解の正値性の強さを顕在化できた.(3)sublinear非線形境界条件を用いた沿岸漁業収穫モデルの解析は多次元有界領域における初の試みであった.このモデルへの解析的アプローチの進展が今後期待できる.

Report

(7 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • Research Products

    (30 results)

All 2023 2022 2021 2020 2019 2018 Other

All Int'l Joint Research (8 results) Journal Article (9 results) (of which Int'l Joint Research: 7 results,  Peer Reviewed: 9 results) Presentation (8 results) (of which Int'l Joint Research: 3 results,  Invited: 5 results) Remarks (5 results)

  • [Int'l Joint Research] コルドバ大学(アルゼンチン)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] Universidad Nacional de Cordoba(アルゼンチン)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] Universidad Nacional de Cordoba(アルゼンチン)

    • Related Report
      2021 Research-status Report
  • [Int'l Joint Research] Universidad Nacional de Cordoba(アルゼンチン)

    • Related Report
      2020 Research-status Report
  • [Int'l Joint Research] Universidad Nacional de Cordoba(アルゼンチン)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Universidad de Santiago de Chile(チリ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] Universidad Nacional de Cordoba(アルゼンチン)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] Universidad de Santiago de Chile(チリ)

    • Related Report
      2018 Research-status Report
  • [Journal Article] Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting2023

    • Author(s)
      Kenichiro Umezu
    • Journal Title

      Nonlinear Analysis: Real World Applications

      Volume: 70 Pages: 103788-103788

    • DOI

      10.1016/j.nonrwa.2022.103788

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary condition2023

    • Author(s)
      Kenichiro Umezu
    • Journal Title

      Journal of Differential Equations

      Volume: 350 Pages: 124-151

    • DOI

      10.1016/j.jde.2022.12.017

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Nonnegative solutions of an indefinite sublinear Robin problem II: local and global exactness results2022

    • Author(s)
      Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu
    • Journal Title

      Israel Journal of Mathematics

      Volume: 247 Issue: 2 Pages: 661-696

    • DOI

      10.1007/s11856-021-2278-y

    • Related Report
      2022 Research-status Report 2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Uniqueness and sign properties of minimizers in a quasilinear indefinite problem2021

    • Author(s)
      Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu
    • Journal Title

      Communications on Pure and Applied Analysis

      Volume: 20 Issue: 6 Pages: 2313-2322

    • DOI

      10.3934/cpaa.2021078

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Uniqueness and positivity issues in a quasilinear indefinite problem2021

    • Author(s)
      Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 60 Issue: 5

    • DOI

      10.1007/s00526-021-02057-8

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Nonnegative solutions of an indefinite sublinear Robin problem I: positivity, exact multiplicity, and existence of a subcontinuum2020

    • Author(s)
      Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu
    • Journal Title

      Annali di Matematica Pura ed Applicata(1923-)

      Volume: 199 Issue: 5 Pages: 2015-2038

    • DOI

      10.1007/s10231-020-00954-x

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Past and recent contributions to indefinite sublinear elliptic problems2020

    • Author(s)
      Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu
    • Journal Title

      Rendiconti dell'Istituto di Matematica dell'Universita di Trieste

      Volume: 52 Pages: 217-241

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A curve of positive solutions for an indefinite sublinear Dirichlet problem2019

    • Author(s)
      Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu
    • Journal Title

      Discrete and Continuous Dynamical Systems-A

      Volume: 40 Issue: 2 Pages: 617-645

    • DOI

      10.3934/dcds.2020063

    • Related Report
      2019 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Loop type subcontinua of positive solutions for indefinite concave-convex problems2019

    • Author(s)
      Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu
    • Journal Title

      Advanced Nonlinear Studies

      Volume: 19 Issue: 2 Pages: 391-412

    • DOI

      10.1515/ans-2018-2027

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Positive solutions for some sublinear Robin problem with an indefinite weight2023

    • Author(s)
      梅津健一郎
    • Organizer
      茨城大学金曜セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] An exact multiplicity result for some sublinear Robin problem with an indefinite weight2023

    • Author(s)
      Kenichiro Umezu
    • Organizer
      The 13th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Logistic elliptic equations with the nonlinear boundary condition arising in coast fishery harvesting2022

    • Author(s)
      梅津健一郎
    • Organizer
      東北大学応用数理解析セミナー
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Uniqueness of a positive solution for the Laplace equation with indefinite superlinear boundary conditions: near a critical case2021

    • Author(s)
      梅津健一郎
    • Organizer
      南大阪応用数学セミナー
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Global exact multiplicity of positive solutions for an indenite sublinear Robin problem2020

    • Author(s)
      梅津健一郎
    • Organizer
      日本数学会2020年年会函数方程式論分科会一般講演
    • Related Report
      2019 Research-status Report
  • [Presentation] Existence of a loop of positive solutions for concave-convex problems with indefinite weights2019

    • Author(s)
      Kenichiro Umezu
    • Organizer
      Equadiff 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Exact multiplicity of positive solutions for an indefinite concave Robin bvp2019

    • Author(s)
      梅津健一郎
    • Organizer
      日本数学会2019年年会函数方程式論分科会
    • Related Report
      2018 Research-status Report
  • [Presentation] Positivity of bifurcating nontrivial nonnegative solutions of indefinite concave problems2018

    • Author(s)
      Kenichiro Umezu
    • Organizer
      The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Remarks] 茨城大学研究者情報総覧 梅津健一郎

    • URL

      https://info.ibaraki.ac.jp/Profiles/17/0001645/profile.html

    • Related Report
      2023 Annual Research Report 2022 Research-status Report 2021 Research-status Report
  • [Remarks] 論文: 茨城大学研究者情報総覧(梅津健一郎)

    • URL

      https://info.ibaraki.ac.jp/Profiles/17/0001645/detail.html?lang=ja&achievement=ronbun

    • Related Report
      2020 Research-status Report 2019 Research-status Report
  • [Remarks] 研究発表: 茨城大学研究者情報総覧(梅津健一郎)

    • URL

      https://info.ibaraki.ac.jp/Profiles/17/0001645/detail.html?lang=ja&achievement=meeting_achievement

    • Related Report
      2020 Research-status Report 2019 Research-status Report
  • [Remarks] 論文

    • URL

      https://info.ibaraki.ac.jp/Profiles/17/0001645/theses1.html

    • Related Report
      2018 Research-status Report
  • [Remarks] 研究発表

    • URL

      https://info.ibaraki.ac.jp/Profiles/17/0001645/meeting_achieve1.html

    • Related Report
      2018 Research-status Report

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Published: 2018-04-23   Modified: 2025-01-30  

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