East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

BOUNDARY VALUE PROBLEM FOR ONE-DIMENSIONAL ELLIPTIC JUMPING PROBLEM WITH CROSSING n-EIGENVALUES

  • JUNG, TACKSUN (Department of Mathematics, Kunsan National University) ;
  • CHOI, Q-HEUNG (Department of Mathematics Education, Inha University)
  • Received : 2018.10.20
  • Accepted : 2018.12.04
  • Published : 2019.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

This paper is dealt with one-dimensional elliptic jumping problem with nonlinearities crossing n eigenvalues. We get one theorem which shows multiplicity results for solutions of one-dimensional elliptic boundary value problem with jumping nonlinearities. This theorem is that there exist at least two solutions when nonlinearities crossing odd eigenvalues, at least three solutions when nonlinearities crossing even eigenvalues, exactly one solutions and no solution depending on the source term. We obtain these results by the eigenvalues and the corresponding normalized eigenfunctions of the elliptic eigenvalue problem and Leray-Schauder degree theory.

Keywords

References

  1. RA. Adams and JJF. Fournier, Sobolev spaces, 2nd ed, Academic Press, 2003.
  2. Q.H. Choi and T. Jung, A nonlinear suspension bridge equation with nonconstant load, Nonlinear Analysis TMA. 35, 649-668 (1999). https://doi.org/10.1016/S0362-546X(97)00616-0
  3. Q.H. Choi and T. Jung, An application of a variational reduction method to a nonlinear wave equation, J. Diff. Eq. 117, 390-410 (1995). https://doi.org/10.1006/jdeq.1995.1058
  4. Q.H. Choi and T. Jung, Multiplicity results for the nonlinear suspension bridge equation, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, Vol.9, 29-38 (2002).
  5. Q.H. Choi and T. Jung and P. J. McKenna, The study of a nonlinear suspension bridge equation by a variational reduction method, Applicable Analysis, 50, 73-92 (1993). https://doi.org/10.1080/00036819308840185
  6. LC. Evans, Partial differential equations, 2nd ed. Providence R1; American Mathematical Society, 2010.
  7. M. Ghergu and V. Radulescu, ingular elliptic problems, bifurcation and asymptotic analysis, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press,
  8. A. Le, Eigenvalue Problems for the p-Laplacian, Nonlinear Analysis 64 1057-1099, (2006). https://doi.org/10.1016/j.na.2005.05.056
  9. R. Manasevich and J. Mawhin, Periodic solutions for nonlinear systems with p-Laplacian- like operators, J. Differential Equations 145, 367-393 (1998). https://doi.org/10.1006/jdeq.1998.3425
  10. R. Manasevich and J. Mawhin, Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators, J. Korean Math. Society 37, 665-685 (2000).
  11. P.J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rat. Mech. Anal. 98, 167-177, (1987). https://doi.org/10.1007/BF00251232
  12. D. O'Regan, Some general existence principles and results for $({\phi}(y^{\prime}))^{\prime}=qf(t,\;y,\;y^{\prime})$, 0 < t < 1, SIAM J. Math. Anal., 24, 648-668 (1993). https://doi.org/10.1137/0524040
  13. M. del Pino, M. Elgueta and R. Manasevich, A homotopic deformation along p of a LeraySchauder degree result and existence for ($(\left|u^{\prime}\right|^{p-2}u^{\prime}){^{\prime}}$ + f(x, u) = 0, u(0) = u(T) = 0, p > 1, J. Differential Equations 80, 1-13 (1898). https://doi.org/10.1016/0022-0396(89)90093-4