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2011 Fiscal Year Final Research Report

A comprehensive study of nonlinear problems via variational approaches

Research Project

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Project/Area Number 20340037
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionWaseda University

Principal Investigator

TANAKA Kazunaga  早稲田大学, 理工学術院, 教授 (20188288)

Co-Investigator(Kenkyū-buntansha) OZAWA Tohru  早稲田大学, 理工学術院, 教授 (70204196)
OTANI Mitsuharu  早稲田大学, 理工学術院, 教授 (30119656)
NISHIDA Takaaki  早稲田大学, 理工学術院, 教授 (70026110)
YAMAZAKI Masao  早稲田大学, 理工学術院, 教授 (20174659)
YAMADA Yoshio  早稲田大学, 理工学術院, 教授 (20111825)
YANAGIDA Eiji  東京工業大学, 理工学研究科, 教授 (80174548)
KURATA Kazuhiro  首都大学東京, 理工学研究科, 教授 (10186489)
ADACHI Shinji  静岡大学, 工学部, 准教授 (40339685)
HIRATA Jun  早稲田大学, 理工学術院, 助手 (10580483)
SEKIGUCHI Masayoshi  木更津工業高等専門学校, 基礎学系, 教授 (30236088)
Project Period (FY) 2008 – 2011
Keywords変分問題 / 非線型楕円型方程式 / 特異摂動問題 / ハミルトン系
Research Abstract

We study nonlinear problems via variational approaches. Especially (1) we study singular perturbation problems for nonlinear Schrodinger equations and systems. We introduce a new purely variational method which enables us to construct concentrating solutions in a very general setting. (2) We study nonlinear elliptic equations and systems in various settings. We give a new variational construction of radially symmetric ground states. We also study stability and instability of solutions. (3) We also study highly oscillatory solutions in 1-dimensional singular perturbation problems. We give characterization and existence result.

  • Research Products

    (49 results)

All 2012 2011 2010 2009 2008 Other

All Journal Article (37 results) (of which Peer Reviewed: 37 results) Presentation (12 results)

  • [Journal Article] Multiple solutions for semilinear elliptic equations with nonlinear boundary conditions2012

    • Author(s)
      J. Harada, M. Otani
    • Journal Title

      Electron. J. Diff. Eq.

      Volume: No. 33 Pages: 9

    • Peer Reviewed
  • [Journal Article] On limit systems for some population models with cross-diffusion2012

    • Author(s)
      K. Kuto, Y. Yamada
    • Journal Title

      Discrete Contin. Dyn. Systems Series B

      Volume: 17 Pages: 2745-2769

    • Peer Reviewed
  • [Journal Article] Asymptotic properties of ground states of quasilinear Schrodinger equations with H^1-subcritical exponent2012

    • Author(s)
      S. Adachi, T. Watanabe
    • Journal Title

      Adv. Nonlinear Stud.

      Volume: 12 Pages: 255-279

    • Peer Reviewed
  • [Journal Article] Uniqueness of the ground state solutions of quasilinear Schrodinger equations2012

    • Author(s)
      S. Adachi, T. Watanabe
    • Journal Title

      Nonlinear Anal.

      Volume: 75, no. 2 Pages: 819-833

    • Peer Reviewed
  • [Journal Article] A local mountain pass type result for a system of nonlinear Schrodinger equations2011

    • Author(s)
      N. Ikoma, K. Tanaka
    • Journal Title

      Calc. Var. Partial Differential Equations

      Volume: 40 Pages: 449-480

    • Peer Reviewed
  • [Journal Article] Invariant elliptic estimates2011

    • Author(s)
      Y. Cho, T. Ozawa, Y.-S. Shim
    • Journal Title

      J. Math. Anal. Appl.

      Volume: 382 Pages: 162-171

    • Peer Reviewed
  • [Journal Article] H^2-solutions for some elliptic equations with nonlinear boundary conditions.2011

    • Author(s)
      J. Harada, S. Hashimoto, M. Otani
    • Journal Title

      Adv. Math. Sci. Appl.

      Volume: 21, no. 2 Pages: 435-446

    • Peer Reviewed
  • [Journal Article] Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium2011

    • Author(s)
      M. A. Efendiev, M. Otani
    • Journal Title

      Ann. Inst. H. Poincare Anal. Non Lineaire

      Volume: 28 Pages: 565-582

    • Peer Reviewed
  • [Journal Article] A free boundary problem for a reaction-diffusion equation appearing in ecology2011

    • Author(s)
      Y. Kaneko, Y. Yamada
    • Journal Title

      Adv. Math. Sci. Appl.

      Volume: 21 Pages: 467-492

    • Peer Reviewed
  • [Journal Article] G-invariant positive solutions for a quasilinear Schodinger equation2011

    • Author(s)
      S. Adachi, T. Watanabe
    • Journal Title

      Adv. Differential Equations

      Volume: 16, no. 3-4 Pages: 289-324

    • Peer Reviewed
  • [Journal Article] Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation2011

    • Author(s)
      K. Kurata, K. Morimoto
    • Journal Title

      Disc. Contin. Dyn. Syst.

      Volume: 31 Pages: 139-164

    • Peer Reviewed
  • [Journal Article] Positive solutions for Lotka-Volterra competition systems with large cross-diffusion2010

    • Author(s)
      K. Kuto, Y. Yamada
    • Journal Title

      Appl. Anal.

      Volume: 89 Pages: 1037-1066

    • Peer Reviewed
  • [Journal Article] Nonlinear scalar field equations in R^N: mountain pass and symmetric mountain pass approaches2010

    • Author(s)
      J. Hirata, N. Ikoma, K. Tanaka
    • Journal Title

      Topol. Methods Nonlinear Anal.

      Volume: 35 Pages: 253-276

    • Peer Reviewed
  • [Journal Article] Sign-changing multi-bump solutions for nonlinear Schrodinger equations with steep potential wells2009

    • Author(s)
      Y. Sato, K. Tanaka
    • Journal Title

      Trans. Amer. Math. Soc.

      Volume: 361 Pages: 6205-6253

    • Peer Reviewed
  • [Journal Article] Elliptic estimates independent of domain expansion2009

    • Author(s)
      Y. Cho, T. Ozawa, Y.-S. Shim
    • Journal Title

      Calc. Var. PDE

      Volume: 34 Pages: 321-339

    • Peer Reviewed
  • [Journal Article] Sobolev inequalities with symmetry2009

    • Author(s)
      Y. Cho, T. Ozawa
    • Journal Title

      Commun. Contemp. Math.

      Volume: 11, no. 3 Pages: 355-365

    • Peer Reviewed
  • [Journal Article] Inequalities associated with dilations2009

    • Author(s)
      T. Ozawa, H. Sasaki
    • Journal Title

      Commun. Contemp. Math.

      Volume: 11 Pages: 265-277

    • Peer Reviewed
  • [Journal Article] Semilinear elliptic equations with nonlinear boundary conditions2009

    • Author(s)
      J. Harada, M. Otani
    • Journal Title

      Nonlinear Anal.

      Volume: 71 Pages: e2965-e2968

    • Peer Reviewed
  • [Journal Article] H^2-solutions for some elliptic equations with nonlinear boundary conditions2009

    • Author(s)
      J. Harada, M. Otani
    • Journal Title

      Discrete Contin. Dyn. Syst. 2009, Dyn. Sys. Diff. Eq.Appl. 7th AIMS Conference, suppl.

      Pages: 333-339

    • Peer Reviewed
  • [Journal Article] Global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion2009

    • Author(s)
      Y. Yamada
    • Journal Title

      Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, edited by Yihong Du, Hitoshi Ishii and Wei-Yueh Lin

      Volume: World Scientific Publ.,Hackensack, NJ Pages: 282-299

    • Peer Reviewed
  • [Journal Article] Limiting characterization of stationary solutions for a prey-predator model with nonlinear diffusion of fractional type2009

    • Author(s)
      K. Kuto, Y. Yamada
    • Journal Title

      Diff. Integral Eq.

      Volume: 22 Pages: 725-752

    • Peer Reviewed
  • [Journal Article] On the long-time limit of positive solutions to the degenerate logistic equation2009

    • Author(s)
      Y. Du, Y. Yamada
    • Journal Title

      Discrete Contin. Dyn. Syst.

      Volume: 25 Pages: 123-132

    • Peer Reviewed
  • [Journal Article] Hardy type inequality and application to the stability of degenerate stationary waves2009

    • Author(s)
      S. Kawashima, K. Kurata
    • Journal Title

      J. Funct. Anal.

      Volume: 257 Pages: 1-19

    • Peer Reviewed
  • [Journal Article] Sign-changing multi-peak solutions for nonlinear Schrodinger equations with critical frequency2008

    • Author(s)
      Y. Sato
    • Journal Title

      Commun. Pure Appl. Anal.

      Volume: 7, no. 4 Pages: 883-903

    • Peer Reviewed
  • [Journal Article] Uniqueness of radially symmetric positive solutions for -Δu+u=u^p in an annulus2008

    • Author(s)
      P. Felmer, S. Martinez, K. Tanaka
    • Journal Title

      J. Diff. Eq.

      Volume: 245 Pages: 1198-1209

    • Peer Reviewed
  • [Journal Article] Standing waves for nonlinear Schrodinger equations with a general nonlinearity: one and two dimensional cases2008

    • Author(s)
      J. Byeon, L. Jeanjean, K. Tanaka
    • Journal Title

      Comm. Partial Diff. Eq.

      Volume: 33 Pages: 1113-1136

    • Peer Reviewed
  • [Journal Article] Highly oscillatory behavior of the activator in the Gierer and Meinhardt system2008

    • Author(s)
      P. Felmer, S. Martinez, K. Tanaka
    • Journal Title

      Math. Ann.

      Volume: 340 Pages: 749-773

    • Peer Reviewed
  • [Journal Article] Positive solutions for Lotka-Volterra Systems with cross-diffusion2008

    • Author(s)
      Y. Yamada
    • Journal Title

      Handbook of Differential Equations: Stationary Partial Differential Equations, Edited by M.Chipot, Elsevier/North-Holland, Amsterdam

      Volume: 6 Pages: 411-501

    • Peer Reviewed
  • [Journal Article] Nonlinear Schrodinger equation with a point defect2008

    • Author(s)
      R. Fukuizumi, M. Ohta, T. Ozawa
    • Journal Title

      Ann. Inst. H. Poincare Anal. Non Lineaire

      Volume: 25 Pages: 837- 845

    • Peer Reviewed
  • [Journal Article] Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation2008

    • Author(s)
      K. Kurata, K. Morimoto
    • Journal Title

      Commun. Pure Appl. Anal.

      Volume: 7 Pages: 1443-1482

    • Peer Reviewed
  • [Journal Article] Compact embedding from W^<1,2>(Ω) to L^<q(x)>(Ω) and its application to nonlinear elliptic boundary value problem with variable critical exponent2008

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

      J. Math. Anal. Appl.

      Volume: 339 Pages: 1386-1394

    • Peer Reviewed
  • [Journal Article] Optimal spatial harvesting strategy and symmetry-breaking2008

    • Author(s)
      K. Kurata, J. Shi
    • Journal Title

      Appl. Math. Optim.

      Volume: 58, no. 1 Pages: 89-110

    • Peer Reviewed
  • [Journal Article] Semiclassical standing waves with clustering peaks for nonlinear Schrodinger equations

    • Author(s)
      J. Byeon, K. Tanaka
    • Journal Title

      Memoir Amer. Math. Soc.

      Volume: (掲載予定)

    • Peer Reviewed
  • [Journal Article] Semi-classical standing waves for nonlinear Schrodinger equations at structurally stable critical points of the potential

    • Author(s)
      J. Byeon, K. Tanaka
    • Journal Title

      J. Eur. Math. Soc.

      Volume: (掲載予定)

    • Peer Reviewed
  • [Journal Article] Nonlinear elliptic equations in strip-like domains

    • Author(s)
      J. Byeon, K. Tanaka
    • Journal Title

      Advanced Nonlinear Studies

      Volume: (掲載予定)

    • Peer Reviewed
  • [Journal Article] On a system of nonlinear Schrodinger equations with quadratic interaction

    • Author(s)
      N. Hayashi, T. Ozawa, K. Tanaka
    • Journal Title

      Annal. Inst. H. Poincare, Anal. non lineaire

      Volume: (掲載予定)

    • Peer Reviewed
  • [Journal Article] On the multiple existence of semi-positive solutions for a nonlinear Schrodinger system

    • Author(s)
      Z.-Q. Wang, Y. Sato
    • Journal Title

      Ann. Inst. H. Poincare Anal. Non Lineaire

      Volume: (掲載予定)

    • Peer Reviewed
  • [Presentation] 連立非線型楕円型方程式の半正値解の多重存在について2012

    • Author(s)
      佐藤洋平
    • Organizer
      第4回福島楕円型・放物型微分方程式研究集会
    • Place of Presentation
      福島大学
    • Year and Date
      2012-01-28
  • [Presentation] On Brinkman-Forchheimer equaitons of flow in double-diffusive convection2011

    • Author(s)
      M. Otani
    • Organizer
      Dutch-Japanese workshop
    • Place of Presentation
      アイントホーフェン, オランダ.
    • Year and Date
      2011-11-08
  • [Presentation] A sharp bilinear estimate for Klein-Gordon equation, Evolution equations, related topics and application2011

    • Author(s)
      T. Ozawa
    • Place of Presentation
      早稲田大学
    • Year and Date
      2011-03-23
  • [Presentation] On limit systems for some population models with cross-diffusion2010

    • Author(s)
      Y. Yamada
    • Organizer
      Workshop on PDE Models of Biological Processes
    • Place of Presentation
      National Center of Theoretical Sciences (NCTS), Hsinchu, Taiwan
    • Year and Date
      2010-12-17
  • [Presentation] Positive solutions for a singularly perturbed system of nonlinear Schrodinger equations2010

    • Author(s)
      K.Tanaka
    • Organizer
      Workshop on Variational Methods in Nonlinear Differential Equations
    • Place of Presentation
      Oaxaca, Mexico
    • Year and Date
      2010-10-22
  • [Presentation] Mountain pass and symmetric mountain pass approaches to nonlinear scalar field equations and related problems2010

    • Author(s)
      K.Tanaka
    • Organizer
      Conference On Nonlinear PDE's
    • Place of Presentation
      POSTECH, Korea
    • Year and Date
      2010-10-05
  • [Presentation] Infinite dimensionality of global attractors for some quasilinear parabolic equations2009

    • Author(s)
      M. Otani
    • Organizer
      The second Chile-Japan workshop on nonlinear elliptic and parabolic PDEs
    • Place of Presentation
      明治大学
    • Year and Date
      2009-12-04
  • [Presentation] A local mountain pass type result for a system of nonlinear Schroedinger equations2009

    • Author(s)
      K. Tanaka
    • Organizer
      The second Chile-Japan workshop on nonlinear elliptic and parabolic PDEs
    • Place of Presentation
      明治大学
    • Year and Date
      2009-12-02
  • [Presentation] A local mountain pass type result for a system of nonlinear Schroedinger equations2009

    • Author(s)
      K. Tanaka
    • Organizer
      The first Italian-Japanese workshop on geometric properties for parabolic and elliptic PDEs
    • Place of Presentation
      東北大学
    • Year and Date
      2009-06-18
  • [Presentation] A local mountain pass type result for a system of nonlinear Schrodinger equations2009

    • Author(s)
      K.Tanaka
    • Organizer
      International Conference on Variational Methods
    • Place of Presentation
      Nankai Univeristy, Tianjin, China
    • Year and Date
      2009-05-22
  • [Presentation] Infinite dimensionality of global attractors for parabolic equations governed by p-Laplacian in heterogeneous medium2009

    • Author(s)
      M. Otani
    • Organizer
      Current advances in applied nonlinear analysis and mathematical modeling
    • Place of Presentation
      Warsaw
    • Year and Date
      2009-05-18
  • [Presentation] Limiting behavior of solutions for some reaction-diffusion systems with nonlinear diffusion2008

    • Author(s)
      Y. Yamada
    • Organizer
      World Congress of Nonlinear Analysts 2008
    • Place of Presentation
      Orlando, Florida, USA
    • Year and Date
      2008-07-03

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Published: 2013-07-31   Modified: 2014-11-26  

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