Skip to content

Advertisement

  • Research Article
  • Open Access

Eigenvalue Problems and Bifurcation of Nonhomogeneous Semilinear Elliptic Equations in Exterior Strip Domains

Boundary Value Problems20062007:014731

https://doi.org/10.1155/2007/14731

  • Received: 19 July 2006
  • Accepted: 20 October 2006
  • Published:

Abstract

We consider the following eigenvalue problems: in in where , , is a smooth bounded domain, , is a smooth bounded domain in such that . Under some suitable conditions on and , we show that there exists a positive constant such that the above-mentioned problems have at least two solutions if , a unique positive solution if , and no solution if . We also obtain some bifurcation results of the solutions at .

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Eigenvalue Problem

[12345678910111213141516171819]

Authors’ Affiliations

(1)
Center of General Education, Chang Gung University, Kwei-San, Tao-Yuan, 333, Taiwan

References

  1. Cao DM: Eigenvalue problems and bifurcation of semilinear elliptic equation in . Nonlinear Analysis. Theory, Methods & Applications 1995,24(4):529–554. 10.1016/0362-546X(94)E0071-NMATHMathSciNetView ArticleGoogle Scholar
  2. Zhu XP: A perturbation result on positive entire solutions of a semilinear elliptic equation. Journal of Differential Equations 1991,92(2):163–178. 10.1016/0022-0396(91)90045-BMATHMathSciNetView ArticleGoogle Scholar
  3. Cao DM, Zhou H-S: Multiple positive solutions of nonhomogeneous semilinear elliptic equations in . Proceedings of the Royal Society of Edinburgh. Section A 1996,126(2):443–463. 10.1017/S0308210500022836MATHMathSciNetView ArticleGoogle Scholar
  4. Zhu XP, Zhou HS: Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains. Proceedings of the Royal Society of Edinburgh. Section A 1990,115(3–4):301–318. 10.1017/S0308210500020667MATHMathSciNetView ArticleGoogle Scholar
  5. Esteban MJ: Nonlinear elliptic problems in strip-like domains: symmetry of positive vortex rings. Nonlinear Analysis. Theory, Methods & Applications 1983,7(4):365–379. 10.1016/0362-546X(83)90090-1MATHMathSciNetView ArticleGoogle Scholar
  6. Lions P-L: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1984,1(2):109–145.MATHGoogle Scholar
  7. Lions P-L: The concentration-compactness principle in the calculus of variations. The locally compact case. II. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1984,1(4):223–283.MATHGoogle Scholar
  8. Bahri A, Lions P-L: On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1997,14(3):365–413.MATHMathSciNetView ArticleGoogle Scholar
  9. Lions P-L: On positive solutions of semilinear elliptic equations in unbounded domains. In Nonlinear Diffusion Equations and Their Equilibrium States, II (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ.. Volume 13. Edited by: Ni W-M, Peletier LA, Serrin J. Springer, New York, NY, USA; 1988:85–122.View ArticleGoogle Scholar
  10. Hsu T-S: Exactly two positive solutions of nonhomogeneous semilinear elliptic equations in unbounded cylinder domains. Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 2005,12(5):685–705.MATHMathSciNetGoogle Scholar
  11. Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order, Fundamental Principles of Mathematical Sciences. Volume 224. 2nd edition. Springer, Berlin, Germany; 1983.View ArticleGoogle Scholar
  12. Adams RA: Sobolev Spaces, Pure and Applied Mathematics. Volume 65. Academic Press, New York, NY, USA; 1975.Google Scholar
  13. Hsu T-S: Multiple solutions for semilinear elliptic equations in unbounded cylinder domains. Proceedings of the Royal Society of Edinburgh. Section A 2004,134(4):719–731. 10.1017/S0308210500003449MATHMathSciNetView ArticleGoogle Scholar
  14. Ekeland I: Nonconvex minimization problems. Bulletin of the American Mathematical Society 1979,1(3):443–474. 10.1090/S0273-0979-1979-14595-6MATHMathSciNetView ArticleGoogle Scholar
  15. Graham-Eagle J: Monotone methods for semilinear elliptic equations in unbounded domains. Journal of Mathematical Analysis and Applications 1989,137(1):122–131. 10.1016/0022-247X(89)90276-XMATHMathSciNetView ArticleGoogle Scholar
  16. Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. Journal of Functional Analysis 1973,14(4):349–381. 10.1016/0022-1236(73)90051-7MATHMathSciNetView ArticleGoogle Scholar
  17. Crandall MG, Rabinowitz PH: Bifurcation, perturbation of simple eigenvalues and linearized stability. Archive for Rational Mechanics and Analysis 1973,52(2):161–180.MATHMathSciNetView ArticleGoogle Scholar
  18. Korman P, Li Y, Ouyang T: Exact multiplicity results for boundary value problems with nonlinearities generalising cubic. Proceedings of the Royal Society of Edinburgh. Section A 1996,126(3):599–616. 10.1017/S0308210500022927MATHMathSciNetView ArticleGoogle Scholar
  19. Lien WC, Tzeng SY, Wang HC: Existence of solutions of semilinear elliptic problems on unbounded domains. Differential and Integral Equations 1993,6(6):1281–1298.MATHMathSciNetGoogle Scholar

Copyright

Advertisement