Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지

LERAY-SCHAUDER DEGREE THEORY APPLIED TO THE PERTURBED PARABOLIC PROBLEM

  • Jung, Tacksun (Department of Mathematics Kunsan National University) ;
  • Choi, Q-Heung (Department of Mathematics Education Inha University)
  • Received : 2009.05.20
  • Published : 20090600
Download PDF
⟨ Previous Next ⟩

Abstract

We show the existence of at least four solutions for the perturbed parabolic equation with Dirichlet boundary condition and periodic condition when the nonlinear part cross two eigenvalues of the eigenvalue problem of the Laplace operator with boundary condition. We obtain this result by using the Leray-Schauder degree theory, the finite dimensional reduction method and the geometry of the mapping. The main point is that we restrict ourselves to the real Hilbert space instead of the complex space.

Keywords

Acknowledgement

Supported by : Korea Research Foundation

References

  1. K. C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhauser, (1993)
  2. Q. H. Choi and T. Jung, An application of a variational reduction method to a nonlinear wave equation, J. Differential Equations, 117 (1995), 390-410. https://doi.org/10.1006/jdeq.1995.1058
  3. Q. H. Choi and T. Jung, Multiple solutions for the nonlinear parabolic problem, Journal of the Chungcheong MathematicalSociety, To be be appeared (2009).
  4. Q. H. Choi and T. Jung, Multiple periodic solutions of a semilinear wave equation at double external resonances, Communications in Applied Analysis, 3, no.1 (1999), 73-84.
  5. Q. H. Choi and T. Jung, Multiplicity results for nonlinear wave equations with nonlinearities crossing eigenvalues, Hokkaido Mathematical Journal, 24, no. 1 (1995), 53-62. https://doi.org/10.14492/hokmj/1380892535
  6. A. C. Lazer and McKenna, Some multiplicity results for a class of semilinear elliptic and parabolic boundary value problems, J. Math. Anal. Appl., 107 (1985), 371-395. https://doi.org/10.1016/0022-247X(85)90320-8
  7. A.C. Lazer and McKenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl., 181 (1994), 648-655. https://doi.org/10.1006/jmaa.1994.1049
  8. P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge,Archive for Rational Mechanics and Analysis, 98, no. 2 (1987), 167-177.
  9. P. J. McKenna and W. Walter, On the multiplicity of the solution set of some nonlinear boundary value problems, Nonlinear Analysis TMA, 8, no. 8 (1984), 893-907. https://doi.org/10.1016/0362-546X(84)90110-X
  10. A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Analysis TMA, 31 (1998), 895-908. https://doi.org/10.1016/S0362-546X(97)00446-X
  11. A. M. Micheletti and A. Pistoia, Nontrivial solutions for some fourth order semilinear elliptic problems, Nonlinear Analysis, 34 (1998), 509-523. https://doi.org/10.1016/S0362-546X(97)00596-8
  12. J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach, New York, (1969).
  13. P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations (C.B.M.S. Reg. Conf. Ser. in Math. 6), American Mathematical Society, Providence, R1, (1986).
  14. G. Tarantello, A note on a semilinear elliptic problem, Diff. Integ. Equations. 561-565 (1992).