Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지

AT LEAST FOUR SOLUTIONS TO THE NONLINEAR ELLIPTIC SYSTEM

  • Jung, Tacksun (Department of Mathematics Kunsan National University) ;
  • Choi, Q-Heung (Department of Mathematics Education Inha University)
  • Received : 2009.05.13
  • Published : 20090600
Download PDF
⟨ Previous Next ⟩

Abstract

We prove the existence of multiple solutions (${\xi},{\eta}$) for perturbations of the elliptic system with Dirichlet boundary condition $$(0.1)\;\begin{array}{lcr}A{\xi}+g_1({\xi}+ 2{\eta})=s{\phi}_1+h\;in\;{\Omega},\\A{\xi}+g_2({\xi}+ 2{\eta})=s{\phi}_1+h\;in\;{\Omega},\end{array}$$ where $lim_{u{\rightarrow}{\infty}}\frac{gj(u)}{u}={\beta}_j$, $lim_{u{\rightarrow}-{\infty}}\frac{gj(u)}{u}={\alpha}_j$ are finite and the nonlinearity $g_1+2g_2$ crosses eigenvalues of A.

Keywords

Acknowledgement

Supported by : Kunsan National University

References

  1. H. Amann, Saddle points and multiple solutions of differential equation, Math. Z., 169 (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, S. Chun and T. Jung, The multiplicity of solutions and geometry of the nonlinear elliptic equation, Studia Math., 120 (1996), 259-270. https://doi.org/10.4064/sm-120-3-259-270
  5. 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
  6. A. C. Lazer and P.J. McKenna, Some multiplicity results for a class of semi-linear 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 P.J. McKenna, Critical points theory and boundary value problems with nonlinearities crossing multiple eigenvalues II, Comm.in P. D. E., 11 (1986), 1653-1676. https://doi.org/10.1080/03605308608820479
  8. 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
  9. 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
  10. 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
  11. P. J. McKenna, Topological Methods for Asymmetric Boundary Value Problems, Globa Analysis Research Center, Seoul National University (1993).
  12. 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
  13. 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
  14. Paul H. Rabinobitz, Minimax methods in critical point theory with applications to differential equations, Mathematical Science regional conference series, no. 65, AMS (1984).
  15. G. Tarantello, A note on a semilinear elliptic problem, Diff. Integ. Equations., 5 (1992), 561-565.