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Uniqueness and multiplicity of solutions of nonlinear elliptic problems

Research Project

Project/Area Number 26400160
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Mathematical analysis
Research InstitutionYokohama National University

Principal Investigator

SHIOJI Naoki  横浜国立大学, 大学院工学研究院, 教授 (50215943)

Research Collaborator WATANABE Kohtaro  
Project Period (FY) 2014-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords一意性 / 正値球対称解 / 非退化性 / Pohozaev関数 / 分岐 / 弾性曲線 / 楕円型方程式 / 正値解 / 双曲空間 / 分数べきラプラシアン / 臨界べき
Outline of Final Research Achievements

Using a generalized Pohozaev function, we extend the uniqueness result of positive radial solutions of elliptic equations obtained by Shioji-Watanabe (JDE 255 (2013)). In particular, for an elliptic problem in two dimensional space, we obtained a new uniqueness result, and we show that we can apply it to the Haraux-Weissler equation in two dimensional space. Moreover, for some elliptic problems, we study the nondegeneracy of the unique positive radial solution in a radial function space under almost same assumptions for the uniqueness. We also study its nondegeneracy even in a space which is not restricted to be radial.

Academic Significance and Societal Importance of the Research Achievements

微分方程式を用いてモデルを記述することが多い自然科学や工学において、考えている方程式に対し、解は存在するのか、存在する場合、解は一意なのかあるいは多数存在するのかは重要な問題である。微分方程式論の研究において、解の存在と一意性・多重性は幅広く研究されており、研究成果は微分方程式論の発展のために寄与するものである。

Report

(6 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (22 results)

All 2018 2017 2016 2015 2014

All Journal Article (6 results) (of which Int'l Joint Research: 2 results,  Peer Reviewed: 6 results,  Acknowledgement Compliant: 2 results) Presentation (16 results) (of which Int'l Joint Research: 8 results,  Invited: 3 results)

  • [Journal Article] Rabinowitz Alternative for Non-cooperative Elliptic Systems on Geodesic Balls2018

    • Author(s)
      Rybicki Slawomir、Shioji Naoki、Stefaniak Piotr
    • Journal Title

      Advanced Nonlinear Studies

      Volume: 18 Issue: 4 Pages: 845-862

    • DOI

      10.1515/ans-2018-0012

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Total p-powered curvature of closed curves and flat-core closed p-curves in S^2(G)2018

    • Author(s)
      Naoki Shioji and Kohtaro Watanabe
    • Journal Title

      Communications in Analysis and Geometry

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Uniqueness and nondegeneracy of positive radial solutions of div(ρ∇u)+ρ(-gu+hu^p)=02016

    • Author(s)
      N. Shioji, K. Watanabe
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 55 Issue: 2

    • DOI

      10.1007/s00526-016-0970-2

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] Nonlocal problems at critical growth in contractible domains2015

    • Author(s)
      Sunra Mosconi, Naoki and Marco Squassina
    • Journal Title

      Asymptotic analysis

      Volume: 95 Issue: 1-2 Pages: 79-100

    • DOI

      10.3233/asy-151324

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Uniqueness of positive solutions of Brezis-Nirenberg problems on H^n2015

    • Author(s)
      Naoki Shioji and Kohtaro Watanabe
    • Journal Title

      Linear and nonlinear analysis

      Volume: 1 Pages: 261-270

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Coron problem for fractional equations2015

    • Author(s)
      Simone Secchi, Naoki Shioji, Marco Squassina
    • Journal Title

      Differential Integral Equations

      Volume: 28 Pages: 103-118

    • Related Report
      2014 Research-status Report
    • Peer Reviewed
  • [Presentation] Total p-powered curvature of closed curves2018

    • Author(s)
      Naoki Shioji
    • Organizer
      The 12th AIMS Conference on Dynamical systems, Differential Equations and Applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Uniqueness of positive sokutions of the Brezis-Nirenberg problem in thin sphrical annular domains and its applications2018

    • Author(s)
      Naoki Shioji
    • Organizer
      The 12th AIMS Conference on Dynamical systems, Differential Equations and Applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Total p-powered curvature of closed curves in S^2(G)2018

    • Author(s)
      塩路直樹
    • Organizer
      第35回九州における偏微分方程式研究集会
    • Related Report
      2017 Research-status Report
  • [Presentation] Total p-powered curvature of closed curves in S^2(G)2017

    • Author(s)
      Naoki Shioji
    • Organizer
      International Conference on Topological Nonlinear Analysis
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Uniqueness of positive solutions of the Brezis-Nirenberg problems in thin spherical annular domains and its application2017

    • Author(s)
      Naoki Shioji
    • Organizer
      International Conference on Nonlinear Analysis, PDEs and Applications
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] Uniqueness of positive solutions of Brezis-Nirenberg problems on the hyperbolic space2016

    • Author(s)
      Naoki Shioji
    • Organizer
      The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Orlando (USA)
    • Year and Date
      2016-07-01
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research
  • [Presentation] n次元双曲空間上のBrezis-Nirenberg問題の正値解の一意性について2016

    • Author(s)
      渡辺宏太郎 塩路直樹
    • Organizer
      日本数学会2016年度年会
    • Place of Presentation
      筑波大学
    • Year and Date
      2016-03-16
    • Related Report
      2015 Research-status Report
  • [Presentation] On a Kirchhoff type problem with critical exponent2016

    • Author(s)
      塩路直樹
    • Organizer
      第10回非線形偏微分方程式と変分問題
    • Place of Presentation
      首都大学東京
    • Year and Date
      2016-02-06
    • Related Report
      2015 Research-status Report
  • [Presentation] On a Kirchhoff type problem in R^3 with critical exponent2015

    • Author(s)
      Naoki Shioji
    • Organizer
      VII Symposium on Nonlinear Analysis
    • Place of Presentation
      トルン(ポーランド)
    • Year and Date
      2015-09-14
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] A critical problem for fractional Laplacians in contractible domains2015

    • Author(s)
      Naoki Shioji
    • Organizer
      Equadiff 2015
    • Place of Presentation
      リヨン(フランス)
    • Year and Date
      2015-07-06
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] On closed p-elastic curves on S^2(G)2015

    • Author(s)
      Naoki Shioji
    • Organizer
      2015 International workshop on nonlinear PDE and applications
    • Place of Presentation
      釜山(韓国)
    • Year and Date
      2015-06-11
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] 2次元球面上のL^p弾性閉曲線とflat-core解の存在について2015

    • Author(s)
      渡辺宏太郎・塩路直樹
    • Organizer
      日本数学会2015年度年会
    • Place of Presentation
      明治大学
    • Year and Date
      2015-03-21 – 2015-03-24
    • Related Report
      2014 Research-status Report
  • [Presentation] A critical problem for the fractional laplacian2015

    • Author(s)
      塩路直樹
    • Organizer
      常微分方程式ワークショップ 松山 2015
    • Place of Presentation
      愛媛大学
    • Year and Date
      2015-03-10
    • Related Report
      2014 Research-status Report
  • [Presentation] 2次元球面上のL^p弾性曲線とflat-core解の存在について2014

    • Author(s)
      渡辺宏太郎・塩路直樹
    • Organizer
      RIMS研究集会:実領域における常微分方程式の定性的理論とその応用
    • Place of Presentation
      京都大学
    • Year and Date
      2014-11-04 – 2014-11-06
    • Related Report
      2014 Research-status Report
  • [Presentation] Uniqueness of a positive radial solution of an elliptic problem and its nondegeneracy2014

    • Author(s)
      Naoki Shioji
    • Organizer
      Seoul ICM 2014 Satellite Conference on Variational Methods for Nonlinear Elliptic PDEs
    • Place of Presentation
      Daejon, Korea
    • Year and Date
      2014-08-04 – 2014-08-08
    • Related Report
      2014 Research-status Report
    • Invited
  • [Presentation] Uniqueness of a Positive Radial Solutions of Some Elliptic Problems and Its Nondegeneracy2014

    • Author(s)
      Naoki Shioji
    • Organizer
      The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications
    • Place of Presentation
      Madrid, Spain
    • Year and Date
      2014-07-07 – 2014-07-11
    • Related Report
      2014 Research-status Report

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Published: 2014-04-04   Modified: 2020-03-30  

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