Abstract and Applied Analysis

Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains

Tsing-San Hsu

Full-text: Open access

Abstract

We will show that under suitable conditions on f and h , there exists a positive number λ such that the nonhomogeneous elliptic equation Δ u + u = λ ( f ( x , u ) + h ( x ) ) in Ω , u H 0 1 ( Ω ) , N 2 , has at least two positive solutions if λ ( 0 , λ ) , a unique positive solution if λ = λ , and no positive solution if λ > λ , where Ω is the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.

Article information

Source
Abstr. Appl. Anal., Volume 2007 (2007), Article ID 43018, 19 pages.

Dates
First available in Project Euclid: 27 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1204126586

Digital Object Identifier
doi:10.1155/2007/43018

Mathematical Reviews number (MathSciNet)
MR2283965

Zentralblatt MATH identifier
1157.35371

Citation

Hsu, Tsing-San. Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains. Abstr. Appl. Anal. 2007 (2007), Article ID 43018, 19 pages. doi:10.1155/2007/43018. https://projecteuclid.org/euclid.aaa/1204126586


Export citation

References

  • D. M. Cao, ``Eigenvalue problems and bifurcation of semilinear elliptic equation in $\mathbbR^N$,'' Nonlinear Analysis. Theory, Methods & Applications, vol. 24, no. 4, pp. 529--554, 1995.
  • X. P. Zhu and H. S. Zhou, ``Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains,'' Proceedings of the Royal Society of Edinburgh. Section A, vol. 115, no. 3-4, pp. 301--318, 1990.
  • K.-J. Chen, ``Exactly two entire positive solutions for a class of nonhomogeneous elliptic equations,'' Differential Integral Equations, vol. 17, no. 1-2, pp. 1--16, 2004.
  • T.-S. Hsu, ``Existence of multiple positive solutions of subcritical semilinear elliptic problems in exterior strip domains,'' Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 6, pp. 1203--1228, 2006.
  • R. A. Adams, Sobolev Spaces, vol. 65 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1975.
  • T.-S. Hsu, ``Multiple solutions for semilinear elliptic equations in unbounded cylinder domains,'' Proceedings of the Royal Society of Edinburgh. Section A, vol. 134, no. 4, pp. 719--731, 2004.
  • I. Ekeland, ``Nonconvex minimization problems,'' Bulletin of the American Mathematical Society. New Series, vol. 1, no. 3, pp. 443--474, 1979.
  • H. Amann, ``Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces,'' SIAM Review, vol. 18, no. 4, pp. 620--709, 1976.
  • A. Bahri and P.-L. Lions, ``On the existence of a positive solution of semilinear elliptic equations in unbounded domains,'' Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 14, no. 3, pp. 365--413, 1997.
  • P.-L. Lions, ``On positive solutions of semilinear elliptic equations in unbounded domains,'' in Nonlinear Diffusion Equations and Their Equilibrium States, II (Berkeley, CA, 1986), W. M. Ni, L. A. Peletier, and J. Serrin, Eds., vol. 13 of Math. Sci. Res. Inst. Publ., pp. 85--122, Springer, New York, NY, USA, 1988.
  • A. Ambrosetti and P. H. Rabinowitz, ``Dual variational methods in critical point theory and applications,'' Journal of Functional Analysis, vol. 14, no. 4, pp. 349--381, 1973.