Structure of solution spaces for singular partial differential equations.

• Project/Area Number: 20540197
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Saga University
• Principal Investigator: KAJIKIYA Ryuji 佐賀大学, 工学系・研究科, 教授 (10183261)
• Project Period (FY): 2008 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 関数方程式 / 非線形偏微分方程式 / 劣線形楕円型偏微分方程式 / p-ラプラシアン / 解の分岐 / 特異楕円形偏微分方程式 / 正値解 / 解の正則性 / 固有値問題 / 優線形楕円型偏微分方程式 / 比較定理 / 無限大ラプラス作用素 / 粘性解 / 薄い領域 / 正値解の一意性 / 解の漸近挙動 / 変数分離解 / 解の減衰

Research Abstract

(1) We prove the existence of solutions for semilinear elliptic equations with the singular weight on the boundary. We prove the existence of a smooth positive solution and infinitely many sign-changing solutions.(2) We prove that the bifurcation occurs for one-dimensional p-Laplace equations with singular weights on the boundary.
(3) For degenerate parabolic equations with the infinity Laplacian, we prove that solutions decay as time tends to infinity and investigate the decay order in detail.

Research Products

• [Journal Article] Bifurcation of sign-changing solutions for one-dimensional p-Laplacian with a strong singular weight : p-superlinear at infinity (2011)
  • Author(s): Ryuji Kajikiya, Yong-Hoon Lee and Inbo Sim
  • Journal Title: Nonlinear Analysis, T. M. A. Volume: Vol.74,(No.17) Pages: 5833-5843
  • Peer Reviewed
• [Journal Article] Sublinear elliptic equations with singular coefficients on the boundary (2011)
  • Author(s): Ryuji Kajikiya
  • Journal Title: J. Math. Soc. Japan Volume: Vol.63,(No.1) Pages: 263-294
  • NAID: 10027871371
  • Peer Reviewed
• [Journal Article] Sublinear elliptic equations with singular coefficients on the boundary (2011)
  • Author(s): R.Kajikiya
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 63 Issue: 1 Pages: 263-294
  • DOI: 10.2969/jmsj/06310263
  • NAID: 10027871371
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] Bifurcation of sign-changing solutions for one-dimensional p-Laplacian with a strong singular weight : p-superlinear at infinity (2011)
  • Author(s): R.Kajikiya, Yong-Hoon Lee, Inbo Sim
  • Journal Title: Nonlinear Analysis, T.M.A. Volume: 74 Issue: 17 Pages: 5833-5843
  • DOI: 10.1016/j.na.2011.03.058
  • Peer Reviewed
• [Journal Article] Superlinear elliptic equations with singular coefficients on the boundary (2010)
  • Author(s): Ryuji Kajikiya
  • Journal Title: Nonlinear Analysis, T. M. A. Volume: Vol.73,(No.7) Pages: 2117-2131
  • Peer Reviewed
• [Journal Article] On a criterion for discrete or continuous spectrum of p-Laplace eigenvalue problems with singular sign-changing weights (2010)
  • Author(s): Ryuji Kajikiya, Yong-Hoon Lee and Inbo Sim
  • Journal Title: Nonlinear Analysis, T. M. A. Volume: Vol.72(No.7-8) Pages: 3515-3534
  • Peer Reviewed
• [Journal Article] Superlinear elliptic equations with singular coefficients on the boundary. (2010)
  • Author(s): R.Kajikiya
  • Journal Title: Nonlinear Analysis, T.M.A. 73, (No.7) Pages: 2117-2131
  • Peer Reviewed
• [Journal Article] On a criterion for discrete or continuous spectrum of p-Laplace eigenvalue problems with singular sign-changing weights. (2010)
  • Author(s): R.Kajikiya, Yong-Hoon Lee, Inbo Sim
  • Journal Title: Nonlinear Analysis, T.M.A. Volume: 72 Pages: 3515-3534
  • Peer Reviewed
• [Journal Article] Superlinear elliptic equations with singular coefficients on the boundary. (2010)
  • Author(s): R.Kajikiya
  • Journal Title: Nonlinear Analysis, T.M.A. Volume: 73 Pages: 2117-2131
  • Peer Reviewed
• [Journal Article] Bifurcation of sign-changing solutions for one-dimensional p-Laplacian with a strong singular weight ; p-sublinear at infinity (2009)
  • Author(s): Ryuji Kajikiya, Yong-Hoon Lee and Inbo Sim
  • Journal Title: Nonlinear Analysis, T. M. A. Volume: Vol.71(No.3-4) Pages: 1235-1249
  • Peer Reviewed
• [Journal Article] A priori estimates of positive solutions for sublinear elliptic equations (2009)
  • Author(s): Ryuji Kajikiya
  • Journal Title: Trans. Amer. Math. Soc. Volume: Vol.361(No.7) Pages: 3793-3815
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian (2009)
  • Author(s): G. Akagi, P. Juutinen, R. Kajikiya
  • Journal Title: Math. Ann. Volume: Vol.343(No.4) Pages: 921-953
  • NAID: 120005372395
  • Peer Reviewed
• [Journal Article] A priori estimates of positive solutions for sublinear elliptic equations. (2009)
  • Author(s): R.Kajikiya
  • Journal Title: Trans.Amer.Math.Soc. 361 (No.7) Pages: 3793-3815
  • Peer Reviewed
• [Journal Article] Bifurcation of sign-changing solutions for one-dimensional p-Laplacian with a strong singular weight ; p-sublinear at infinity. (2009)
  • Author(s): R.Kajikiya, Yong-Hoon Lee, Inbo Sim
  • Journal Title: Nonlinear Analysis, T.M.A. 71 (No.3-4) Pages: 1235-1249
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian. (2009)
  • Author(s): G.Akagi, P.Juutinen, R.Kajikiya
  • Journal Title: Math.Ann. 343 (No.4) Pages: 921-953
  • NAID: 120005372395
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian. (2009)
  • Author(s): G.Akagi, P.Juutinen, R.Kajikiya
  • Journal Title: Math.Ann. 343 Pages: 921-953
  • NAID: 120005372395
  • Peer Reviewed
• [Journal Article] A priori estimates of positive solutions for sublinear elliptic equations. (2009)
  • Author(s): R.Kajikiya
  • Journal Title: Transactions of the American Mathematical Society 361 Pages: 3793-3815
  • Peer Reviewed
• [Journal Article] Bifurcation of sign-changing solutions for one-dimensional p-Laplacian with a strong singular weight ; p-sublinear at infinity. (2009)
  • Author(s): R.Kajikiya, Y.-H.Lee, I.Sim
  • Journal Title: Nonlinear Analysis, T.M.A. 71 Pages: 1235-1249
  • Peer Reviewed
• [Journal Article] Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian. (2009)
  • Author(s): G. Akagi, P. Juutinen, R. Kajikiya
  • Journal Title: Math. Ann. 343 Pages: 921-953
  • NAID: 120005372395
  • Peer Reviewed
• [Journal Article] Comparison theorem and uniqueness of positive solutions for sublinear elliptic equations (2008)
  • Author(s): Ryuji Kajikiya
  • Journal Title: Archiv der Mathematik Volume: Vol.91(No.5) Pages: 427-435
  • Peer Reviewed
• [Journal Article] One-dimensional p-Laplacian with a strong singular indefinite weight, I. Eigenvalue (2008)
  • Author(s): Ryuji Kajikiya, Yong-Hoon Lee and Inbo Sim
  • Journal Title: J. Differential Equations Volume: Vol.244(No.8) Pages: 1985-2019
  • Peer Reviewed
• [Journal Article] One-dimensional $p$-Laplacian with a strong singular indefinite weight, I.Eigenvalue. (2008)
  • Author(s): R.Kajikiya, Yong-Hoon Lee, Inbo Sim
  • Journal Title: Journal of Differential Equations 244 (No.8) Pages: 1985-2019
  • Peer Reviewed
• [Journal Article] Comparison theorem and uniqueness of positive solutions for sublinear elliptic equations. (2008)
  • Author(s): R. Kajikiya
  • Journal Title: Archiv der Mathematik 91 Pages: 427-435
  • Peer Reviewed
• [Presentation] Non-even positive solutions of the one dimensional Henon equation (2012)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 東京理科大学
• [Presentation] Least energy solutions and group invariant solutions of the Henon equation (2012)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 東京理科大学
• [Presentation] Symmetry of positive solutions for the generalized Henon equation (2012)
  • Author(s): 梶木屋龍治
  • Organizer: 第4回名古屋微分方程式研究集会
  • Place of Presentation: 名古屋大学
• [Presentation] 2012 International Workshop on Nonlinear PDE and Applications (2012)
  • Author(s): 梶木屋龍治
  • Organizer: Stability and symmetry of positive solutions for the Emden-Fowler equation
  • Place of Presentation: Pusan National University, Pusan, Korea
• [Presentation] International Symposium on Nonlinear Analysis and Optimization 2012 (2012)
  • Author(s): 梶木屋龍治
  • Organizer: Non-radial positive solutions of the Henon equation
  • Place of Presentation: Pukyong National University, Pusan, Korea
• [Presentation] Least energy solutions of the generalized Henon equation (2012)
  • Author(s): 梶木屋龍治
  • Organizer: 松山解析セミナー
  • Place of Presentation: 愛媛大学
• [Presentation] 一般化されたエノン方程式の非球対称解 (2011)
  • Author(s): 梶木屋龍治
  • Organizer: 九州関数方程式セミナー
  • Place of Presentation: 福岡大学セミナーハウス
  • Year and Date: 2011-05-27
• [Presentation] Non-radial positive solutions of the Henon equation (2011)
  • Author(s): 梶木屋龍治
  • Organizer: 第37回発展方程式研究会
  • Place of Presentation: 岐阜大学
• [Presentation] Fast diffusion方程式の解の漸近形の安定性解析(空間N次元の場合) (2011)
  • Author(s): 赤木剛朗, 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 信州大学
• [Presentation] エノン方程式の球対称でない解の存在 (2011)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 信州大学
• [Presentation] エノン方程式の非球対称解の存在について (2011)
  • Author(s): 梶木屋龍治
  • Organizer: 岐阜大学における微分方程式セミナー
  • Place of Presentation: 岐阜大学
• [Presentation] Existence of non-radial positive solutions for the generalized Henon equation (2011)
  • Author(s): 梶木屋龍治
  • Organizer: Loughborough University, Loughborough, England
  • Place of Presentation: Equadiff
• [Presentation] Existence of non-radial solutions for the generalized Henon equation (2011)
  • Author(s): 梶木屋龍治
  • Organizer: 2011 International Workshop on Nonlinear PDE and Applications
  • Place of Presentation: Pusan National University, Pusan, Korea
• [Presentation] Non-even least energy solutions of the Emden-Fowler equation (2011)
  • Author(s): 梶木屋龍治
  • Organizer: 振動理論ワークショップ-徳島2011
  • Place of Presentation: 徳島大学
• [Presentation] Asymptotic behavior of solutions for p-Laplace parabolic equations (2011)
  • Author(s): 梶木屋龍治
  • Organizer: 研究集会「新しい視点からの現象解析と関数方程式」数理解析研究所講究録
  • Place of Presentation: 数理解析研究所
• [Presentation] 放物型pラプラス方程式の解の漸近挙動 (2010)
  • Author(s): 梶木屋龍治
  • Organizer: 九州関数方程式セミナー
  • Place of Presentation: 福岡大学セミナーハウス
  • Year and Date: 2010-05-07
• [Presentation] Asymptotic behavior of solutions for p-Laplace parabolic equations (2010)
  • Author(s): 梶木屋龍治
  • Organizer: The invited lecture
  • Place of Presentation: University of Ulsan, Ulsan, Korea
  • Year and Date: 2010-03-02
• [Presentation] Stability of solutions for fast diffusion equations (2010)
  • Author(s): 梶木屋龍治
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 京都大学
• [Presentation] fast diffusion方程式の解の漸近形の安定性解析 (2010)
  • Author(s): 赤木剛朗, 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 名古屋大学
• [Presentation] Asymptotic behavior of solutions of the porous medium equation (2010)
  • Author(s): 梶木屋龍治
  • Organizer: The invited lecture
  • Place of Presentation: Pusan, Korea
• [Presentation] Asymptotic profiles of sign-changing solutions for parabolic equations associated with p-Laplacian (2010)
  • Author(s): 赤木剛朗, 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 慶應義塾大学
• [Presentation] Bifurcation of a positive solution for one-dimensional p-Laplacian (2010)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 慶應義塾大学
• [Presentation] 1次元pラプラス方程式の正値解の分岐 (2010)
  • Author(s): 梶木屋龍治
  • Organizer: 研究集会「振動理論ワークショップ-倉敷2010」
  • Place of Presentation: 岡山理科大学国際交流センター
• [Presentation] Bifurcation of a positive solution for one-dimensional p-Laplacian. (2010)
  • Author(s): R.Kajikiya
  • Organizer: 慶應義塾大学
  • Place of Presentation: 日本数学会
• [Presentation] Stability of solutions for fast diffusion equations. (2010)
  • Author(s): 梶木屋龍治
  • Organizer: 研究集会「微分方程式の総合的研究」
  • Place of Presentation: 京都大学
• [Presentation] 順序バナッハ空間における峠の定理とその応用 (2009)
  • Author(s): 梶木屋龍治
  • Organizer: 早稲田大学セミナー
  • Place of Presentation: 早稲田大学
  • Year and Date: 2009-11-11
• [Presentation] 薄い領域における劣線形楕円型方程式 (2009)
  • Author(s): 梶木屋龍治
  • Organizer: 熊本大学応用解析セミナー
  • Place of Presentation: 熊本大学
  • Year and Date: 2009-05-09
• [Presentation] ある非線形放物型方程式の解の漸近形について (2009)
  • Author(s): 梶木屋龍治
  • Organizer: 第35回発展方程式研究会,赤木剛朗氏との共同講演
  • Place of Presentation: 中央大学
• [Presentation] Eigenvalues for p-Laplacian problems with a singular sign-changing weight (2009)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 大阪大学
• [Presentation] 順序バナッハ空間における峠の定理とその応用 (2009)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 大阪大学
• [Presentation] 順序バナッハ空間における峠の定理とその応用 (2009)
  • Author(s): 梶木屋龍治
  • Organizer: 研究集会「第5回非線型の諸問題」
  • Place of Presentation: 長崎商工会議所
• [Presentation] 境界に特異性を持つp-Laplacianの固有値問題 (2009)
  • Author(s): 梶木屋龍治
  • Organizer: 研究集会「広島大学における微分方程式セミナー」
  • Place of Presentation: 広島大学
• [Presentation] Mountain pass theorem in ordered Banach spaces and its applications (2009)
  • Author(s): 梶木屋龍治
  • Organizer: Equadidff 12
  • Place of Presentation: Masaryk University, Brno, Czech Republic
• [Presentation] Mountain pass theorem in ordered Banach spaces and its applications (2009)
  • Author(s): 梶木屋龍治
  • Organizer: International Workshop on Nonlinear PDE and Applications
  • Place of Presentation: Pusan National University, Pusan, Korea
• [Presentation] Mountain pass theorem in ordered Banach spaces and its applications (2009)
  • Author(s): 梶木屋龍治
  • Organizer: 研究集会「変分問題とその周辺」数理解析研究所講究録
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Mountain pass theorem in ordered Banach spaces and its applications. (2009)
  • Author(s): R.Kajikiya
  • Organizer: Equadidff 12.
  • Place of Presentation: Masaryk University, Brno, Czech Republic
• [Presentation] Mountain pass theorem in ordered Banach spaces and its applications. (2009)
  • Author(s): R.Kajikiya
  • Organizer: International Workshop on Nonlinear PDE and Applications.
  • Place of Presentation: Pusan National University, Pusan, Korea
• [Presentation] Mountain pass theorem in ordered Banach spaces and its applications. (2009)
  • Author(s): R.Kajikiya
  • Organizer: International Workshop on Nonlinear PDE and Applications.
  • Place of Presentation: Pusan National University, Pusan, Korea,
• [Presentation] Mountain pass theorem in ordered Banach spaces and its applications. (2009)
  • Author(s): R.Kajikiya
  • Organizer: Equadidff 12.
  • Place of Presentation: Masaryk University, Brno, Czech Republic,
• [Presentation] 順序バナッハ空間における峠の定理とその応用. (2009)
  • Author(s): R.Kajikiya
  • Organizer: 日本数学会
  • Place of Presentation: 大阪大学
• [Presentation] Eigenvalues for p-Laplacian problems with a singular sign-changing weight. (2009)
  • Author(s): R.Kajikiya
  • Organizer: 日本数学会
  • Place of Presentation: 大阪大学
• [Presentation] Mountain pass theorem in ordered Banach spaces and its applications (2009)
  • Author(s): R.Kajikiya
  • Organizer: 研究集会「変分問題とその周辺」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] 順序バナッハ空間における峠の定理とその応用. (2009)
  • Author(s): R.Kajikiya
  • Organizer: 研究集会「第5回非線型の諸問題」
  • Place of Presentation: 長崎商工会議所
• [Presentation] ある非線形放物型方程式の解の漸近形について (2009)
  • Author(s): G.Akagi, R.Kajikiya
  • Organizer: 第35回発展方程式研究会
  • Place of Presentation: 中央大学
• [Presentation] 境界に特異性を持つ半線形楕円型方程式 (2008)
  • Author(s): 梶木屋龍治
  • Organizer: 九州関数方程式セミナー
  • Place of Presentation: 九州大学
  • Year and Date: 2008-07-04
• [Presentation] 境界に特異性を持つ半線形楕円型方程式 (2008)
  • Author(s): 梶木屋龍治
  • Organizer: 応用解析研究会
  • Place of Presentation: 早稲田大学
  • Year and Date: 2008-04-19
• [Presentation] Asymptotic behavior of solutions for a degenerate parabolic equation with inhomogeneous Dirichlet data (2008)
  • Author(s): 赤木剛朗, P. Juutinen, 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 東京工業大学
• [Presentation] 境界に特異性を持つ優線形楕円型方程式 (2008)
  • Author(s): 梶木屋龍治
  • Organizer: 日本数学会
  • Place of Presentation: 東京工業大学
• [Presentation] Sublinear elliptic equations with singularity on the boundary. Superlinear elliptic equations with singularity on the boundary (2008)
  • Author(s): 梶木屋龍治
  • Organizer: Intensive Lecture Program and International Conference on Nonlinear PDE and Applications
  • Place of Presentation: Pusan National University, Pusan, Korea
• [Presentation] Sublinear elliptic equations with singularity on the boundary. (2008)
  • Author(s): R.Kajikiya
  • Organizer: Intensive Lecture Program Cand International Conference on Nonlinear PDE and Applications.
  • Place of Presentation: Pusan National University, Pusan, Korea
• [Presentation] Sublinear elliptic equations with singularity on the boundary. (2008)
  • Author(s): 梶木屋 龍治
  • Organizer: Intensive Lecture Program and International Conference on Nonlinear PDE and Applications.
  • Place of Presentation: Pusan National University, Pusan, Korea
• [Presentation] Superlinear elliptic equations with singularity on the boundary. (2008)
  • Author(s): 梶木屋 龍治
  • Organizer: Intensive Lecture Program and International Conference on Nonlinear PDE and Applications.
  • Place of Presentation: Pusan National University, Pusan, Korea
• [Presentation] Asymptotic behavior of solutions for a degenerate parabolie equation with inhomogeneous Dirichlet data. (2008)
  • Author(s): 赤木 剛朗
  • Organizer: 日本数学会
  • Place of Presentation: 東京工業大学