Open Access

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

Boundary Value Problems20062007:014731

DOI: 10.1155/2007/14731

Received: 19 July 2006

Accepted: 20 October 2006

Published: 27 December 2006

Abstract

We consider the following eigenvalue problems: https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq1_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq2_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq3_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq6_HTML.gif is a smooth bounded domain, https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq8_HTML.gif is a smooth bounded domain in https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq9_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq10_HTML.gif . Under some suitable conditions on https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq12_HTML.gif , we show that there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq13_HTML.gif such that the above-mentioned problems have at least two solutions if https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq14_HTML.gif , a unique positive solution if https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq15_HTML.gif , and no solution if https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq16_HTML.gif . We also obtain some bifurcation results of the solutions at https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq17_HTML.gif .

[12345678910111213141516171819]

Authors’ Affiliations

(1)
Center of General Education, Chang Gung University

References

  1. Cao DM: Eigenvalue problems and bifurcation of semilinear elliptic equation in https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq18_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2007%2F14731/MediaObjects/13661_2006_Article_633_IEq19_HTML.gif . 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

© Tsing-San Hsu. 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.