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Multiple existence and structure of solutions for semilinear elliptic equations.

Research Project

Project/Area Number 16540179
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionNagasaki Institute of Applied Science

Principal Investigator

KAJIKIYA Ryuji  Nagasaki Institute of Applied Science, Faculty of Engineering, Professor (10183261)

Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,710,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥210,000)
Fiscal Year 2007: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordselliptic equation / variational method / multiple solutions / sublinear ellintic equation / p-Laplace equation / bifurcation of solution / 特異係数関数 / 無限大ラプラシアン / 放物型方程式 / 解の漸近挙動 / 半線形楕円型方程式 / 比較定理 / variational method / symmetric mountain pass lemma / 偶汎関数
Research Abstract

We discover a new critical point theorem related to the symmetric mountain pass lemma. Our theorem asserts that an even functional on a Banach space has a sequence of critical points converging to zero. By applying it to a sublinear elliptic equation, we prove the existence of infinitely many solutions under a very weak condition.
When the, nonlinear term is not odd, we prove that a sublinear elliptic equation has infinitely many solutions. By considering the Lagrangian functional associated with the elliptic equation as a perturbation from an even functional, we use the symmetry of the functional to prove the existence of solutions. The existence of multiple solutions has been studied for the superlinear elliptic equations. However, little is known about the multiple solutions of the sublinear elliptic equations except for our results.
We prove the regularity of solutions for one-dimensional p-Laplace equations with singular coefficients. By using it, we give a necessary and sufficient condition for the existence of eigenvalues. Then we investigate the structure of the bifurcation of solutions for the one-dimensional p-Laplace equations. Moreover, by using the number of zeros of solutions, we study the direction of the bifurcation branch and show the global existence of the bifurcation curve.

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (38 results)

All 2008 2007 2006 2005 2004 Other

All Journal Article (13 results) (of which Peer Reviewed: 5 results) Presentation (25 results)

  • [Journal Article] One-dimensional p-Laplacian with a strong singular indefinite weight,I.Eigenvalue2008

    • Author(s)
      R.Kajikiya, Y.-H.Lee, I.Sim
    • Journal Title

      Journal of Differential Equations 244

      Pages: 1985-2019

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] One-dimensional p-Lapjacian with a strong singular indefinite weight, I.Eigenvalue.2008

    • Author(s)
      R. Kajikiya Y.-H. Lee, I. Sim
    • Journal Title

      Journal of Differential Equations 244(No.8)

      Pages: 1985-2019

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] One-dimensional p-Laplacian with a strong singular indefinite weight,I.Eigenvalue2008

    • Author(s)
      R.Kajikiya, Y-H.Lee, I.Sim
    • Journal Title

      Journal of Differential Equations 244

      Pages: 1985-2019

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Multiple solutions of sublinear Lane-Emden elliptic equations2006

    • Author(s)
      R.Kajikiya
    • Journal Title

      Calc.Var.Partial Differential Equations 26

      Pages: 29-48

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Multiple solutions of sublinear Lane-Emden elliptic equations.2006

    • Author(s)
      R. Kajikiya
    • Journal Title

      Calculus of Variations and Partial Differential Equations 26(No.1)

      Pages: 29-48

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Multiple solutions of sublinear Lane-Emden elliptic equations.2006

    • Author(s)
      Ryuji Kajikiya
    • Journal Title

      Calculus of Variations and Partial Differential Equations 26・1

      Pages: 29-48

    • Related Report
      2006 Annual Research Report 2005 Annual Research Report
  • [Journal Article] A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations2005

    • Author(s)
      R.Kajikiya
    • Journal Title

      Journal of Functional Analysis 225

      Pages: 352-370

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations.2005

    • Author(s)
      R. Kajikiya
    • Journal Title

      J. Funct. Anal. 225(No.2)

      Pages: 352-370

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations.2005

    • Author(s)
      Ryuji Kajikiya
    • Journal Title

      Journal of Functional Analysis 225・2

      Pages: 352-370

    • Related Report
      2005 Annual Research Report
  • [Journal Article] On the semigroup approach to a class of space-dependent porous medium systems2004

    • Author(s)
      R.Kajikiya, S.Oharu, D.Tebbs
    • Journal Title

      Nihonkai Mathematical Journal 15

      Pages: 45-73

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] On the semigroup approach to a class of space-dependent porous medium systems.2004

    • Author(s)
      R. Kajikiya, S. Oharu, D.
    • Journal Title

      Nihonkai Mathematical Journal 15(No.1)

      Pages: 45-73

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On the semigroup approach to a class of space-dependent porous medium systems.2004

    • Author(s)
      R.Kajikiya, S.Oharu, D.Tebbs
    • Journal Title

      Nihonkai Mathematical Journal 15・1

      Pages: 45-73

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Multiple solutions of sublinear Lane-Emden elliptic equations.

    • Author(s)
      Ryuji Kajikiya
    • Journal Title

      Calculus of Variations and Partial Differential Equations (発表予定)

    • Related Report
      2004 Annual Research Report
  • [Presentation] 境界に特異性を持つ劣線形楕円型方程式2008

    • Author(s)
      梶木屋 龍治
    • Organizer
      日本数学会
    • Place of Presentation
      近畿大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Sublinear elliptic equations with singularity on the boundary.2008

    • Author(s)
      R. Kajikiya
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Kinki Univ
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Asymptotic behavior of solutions for a parabolic equation with the infinity-Laplacian2007

    • Author(s)
      梶木屋 龍治
    • Organizer
      2007 Intersive lecture program and international conference on P.D.E.
    • Place of Presentation
      Pusan National University
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 境界に特異性を持つ1次元pラプラス方程式の解の分岐2007

    • Author(s)
      梶木屋 龍治
    • Organizer
      第33回発展方程式研究会
    • Place of Presentation
      中央大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] ある退化放物型方程式の粘性解の時間大域的挙動について2007

    • Author(s)
      赤木 剛朗(共同講演)
    • Organizer
      第33回発展方程式研究会
    • Place of Presentation
      中央大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 境界に特異性を持つ1次元pラプラス方程式の解の分岐2007

    • Author(s)
      梶木屋 龍治
    • Organizer
      日本数学会
    • Place of Presentation
      東北大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Asymptotic behavior of viscosity solutions for a parabolic equation involving the infinity-Laplacian2007

    • Author(s)
      赤木 剛朗(共同講演)
    • Organizer
      日本数学会
    • Place of Presentation
      東北大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Asymptotic behavior of solutions for a parabolic equation with the infinity-Laplacian.2007

    • Author(s)
      R. Kajikiya
    • Organizer
      2007Intensive lecture program and international conference on P.D.E.
    • Place of Presentation
      Pusan National University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Bifurcation of solutions for one-dimensionalp-Laplace equation with singular coefficients.2007

    • Author(s)
      R. Kajikiya
    • Organizer
      Seminar on evolution equations 33
    • Place of Presentation
      Chuo Univ
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Asymptotic behavior of viscosity solutions for a certain degenerate parabolic equation.2007

    • Author(s)
      G. Akagi
    • Organizer
      Seminar on evolution equations 33
    • Place of Presentation
      Chuo Univ
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Bifurcation of solutions for one-dimensionalp-Laplace equations with singularity on the boundary.2007

    • Author(s)
      R. Kajikiya
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Tohoku Univ
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Asymptotic behavior of viscosity solutions for a parabolic equation involving the infinity-Laplacian2007

    • Author(s)
      G. Akagi
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Tohoku Univ
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Asymptotic behavior of viscosity solutiolls for aparabolic equation involving the infinity'-Laplacian2007

    • Author(s)
      赤木 剛朗
    • Organizer
      日本数学会
    • Place of Presentation
      東北大学
    • Related Report
      2007 Annual Research Report
  • [Presentation] Uniqueness and a priori estimate of positive solutions for sublinear elliptic equations2006

    • Author(s)
      梶木屋 龍治
    • Organizer
      International Workshop on Sublinear Partial Differential Equations.
    • Place of Presentation
      Pusan National University
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 劣線形楕円型方程式の正値解に対する比較定理とアプリオリ評価2006

    • Author(s)
      梶木屋 龍治
    • Organizer
      日本数学会
    • Place of Presentation
      大阪市立大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Uniqueness and a priori estimate of positive solutions for sublinear elliptic equations.2006

    • Author(s)
      R. Kajikiya
    • Organizer
      International Workshop on Sub -linear Partial Differential Equations
    • Place of Presentation
      Pusan National University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Comparison theorem and a priori estimate for positive solutions of sublinear elliptic equations.2006

    • Author(s)
      R. Kajikiya
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Osaka City Univ
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Symmetric mountain pass lemma and multiple solutions of sublinear elliptic equations2005

    • Author(s)
      梶木屋 龍治
    • Organizer
      International conference on differential equations
    • Place of Presentation
      Comenius University
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Symmetric mountain pass lemma and multiple solutions of sublinear elliptic equations.2005

    • Author(s)
      R. Kajikiya
    • Organizer
      International conference on differential equations
    • Place of Presentation
      Comenius University
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Infinitely many small solutions of sublinear elliptic equations2004

    • Author(s)
      梶木屋 龍治
    • Organizer
      The Fourth World Congress of Nonlinear Analysts
    • Place of Presentation
      Orlando,Florida,USA
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A critical point theorem and its applications to sublinear elliptic equations2004

    • Author(s)
      梶木屋 龍治
    • Organizer
      日本数学会
    • Place of Presentation
      北海道大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Symmetric mountain pass lemma and multiple solutions of sublinear elliptic equations2004

    • Author(s)
      梶木屋 龍治
    • Organizer
      第30回発展方程式研究会
    • Place of Presentation
      中央大学
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Infinitely many small solutions of sublinear elliptic equations.2004

    • Author(s)
      R. Kajikiya
    • Organizer
      The Fourth World Congress of Nonlinear Analysts
    • Place of Presentation
      Orlando, Florida, USA
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] A critical point theorem and its applications to sublinear elliptic equations.2004

    • Author(s)
      R. Kajikiya
    • Organizer
      Mathematical Society of Japan
    • Place of Presentation
      Hokkaido Univ
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Symmetric mountain pass lemma and multiple solutions of sublinear elliptic equations.2004

    • Author(s)
      R. Kajikiya
    • Organizer
      Seminar on evolution equations 30
    • Place of Presentation
      Chuo Univ
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2004-04-01   Modified: 2016-04-21  

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