Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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LIMIT RELATIVE CATEGORY THEORY APPLIED TO THE CRITICAL POINT THEORY

  • Jung, Tack-Sun (DEPARTMENT OF MATHEMATICS KUNSAN NATIONAL UNIVERSITY) ;
  • Choi, Q-Heung (DEPARTMENT OF MATHEMATICS EDUCATION INHA UNIVERSITY)
  • Published : 2009.03.31
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Abstract

Let H be a Hilbert space which is the direct sum of five closed subspaces $X_0,\;X_1,\;X_2,\;X_3$ and $X_4$ with $X_1,\;X_2,\;X_3$ of finite dimension. Let J be a $C^{1,1}$ functional defined on H with J(0) = 0. We show the existence of at least four nontrivial critical points when the sublevels of J (the torus with three holes and sphere) link and the functional J satisfies sup-inf variational inequality on the linking subspaces, and the functional J satisfies $(P.S.)^*_c$ condition and $f|X_0{\otimes}X_4$ has no critical point with level c. For the proof of main theorem we use the nonsmooth version of the classical deformation lemma and the limit relative category theory.

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References

  1. T. Bartsch and M. Klapp, Critical point theory for indefinite functionals with symmetries, J. Funct. Anal. 138 (1996), no. 1, 107–136
  2. M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977
  3. K. C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications, 6. Birkhauser Boston, Inc., Boston, MA, 1993
  4. Q. H. Choi, T. Jung, and P. J. McKenna, The study of a nonlinear suspension bridge equation by a variational reduction method, Appl. Anal. 50 (1993), no. 1, 73–92 https://doi.org/10.1080/00036819308840185
  5. M. Degiovanni, Homotopical properties of a class of nonsmooth functions, Ann. Mat. Pura Appl. (4) 156 (1990), 37–71 https://doi.org/10.1007/BF01766973
  6. M. Degiovanni, A. Marino, and M. Tosques, Evolution equations with lack of convexity, Nonlinear Anal. 9 (1985), no. 12, 1401–1443
  7. G. Fournier, D. Lupo, M. Ramos, and M. Willem, Limit relative category and critical point theory, Dynam. Report, 3, 1–23 (1993)