Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지

TWO JUMPING NONLINEAR TERMS AND A NONLINEAR WAVE EQUATION

  • Jung, Tacksun (Department of Mathematics, Kunsan National University) ;
  • Choi, Q-Heung (Department of Mathematics Education, Inha University)
  • Received : 2009.07.30
  • Accepted : 2009.11.06
  • Published : 2009.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

We find the multiple nontrivial solutions of the equation of the form $u_{tt}-u_{xx}=b_1[(u+1)^{+}-1]+b_2[(u+2)^{+}-2]$ with Dirichlet boundary condition. Here we reduce this problem into a two-dimensional problem by using variational reduction method and apply the Mountain Pass theorem to find the nontrivial solutions.

Keywords

References

  1. H. Amann, Saddle points and multiple solutions of differential equation, Math. Z. (1979), 27-166.
  2. A. Ambrosetti and P. H. Rabinowitz, Dual variation methods in critical point theory and applications, J. Functional analysis 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
  3. Q. H. Choi, T. Jung and P. J. McKenna, The study of a nonlinear suspension bridge equation by a variational reduction mehtod, Appl. Anal. 50 (1993), 73-92. https://doi.org/10.1080/00036819308840185
  4. 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
  5. A. C. Lazer and P. J. McKenna, A symmetry theorem and applications to nonlinear partial differential equations, J. Differential Equations 72 (1988), 95-106. https://doi.org/10.1016/0022-0396(88)90150-7
  6. A. C. Lazer and P. J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990), 537-578. https://doi.org/10.1137/1032120
  7. A. C. Lazer and P. J. 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. A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semi-linear elliptic problem, Nonlinear Analysis TMA 31 (1998), 895-908. https://doi.org/10.1016/S0362-546X(97)00446-X
  9. A. M. Micheletti, A.Pistoia, Nontrivial solutions for some fourth order semi-linear elliptic problems, Nonlinear Analysis 34 (1998), 509-523. https://doi.org/10.1016/S0362-546X(97)00596-8
  10. Paul H. Rabinobitz, Minimax methods in critical point theory with applications to differential equations, Regional conference series, No. 65, AMS (1984).
  11. Mathematical Science region G. Tarantello, A note on a semilinear elliptic problem, Diff. Integ. Equations 5 (1992), no. 3, 561-565.