Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T18:10:29.204Z Has data issue: false hasContentIssue false
  • English
  • Français

A perturbation result of semilinear elliptic equations in exterior strip domains

Published online by Cambridge University Press:  14 November 2011

Tsing-san Hsu  and
Hwai-chiuan Wang
Tsing-san Hsu
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan
Hwai-chiuan Wang
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan e-mail: hwang@math.nthu.edu.tw
Get access Rights & Permissions [Opens in a new window]

Synopsis

In this paper we show that if the decay of nonzero ƒ is fast enough, then the perturbation Dirichlet problem −Δu + u = up + ƒ(z) in Ω has at least two positive solutions, where

a bounded C1,1 domain S = × ω Rn, D is a bounded C1,1 domain in Rm+n such that D ⊂⊂ S and Ω = S\D. In case ƒ ≡ 0, we assert that there is a positive higher-energy solution providing that D is small.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 5 , 1997 , pp. 983 - 1004
Copyright
Copyright © Royal Society of Edinburgh 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Adams, R. A.. Sobolev Spaces (New York: Academic Press, 1975).Google Scholar
2Bahri, A. and Berestycki, H.. A perturbation method in critical point theory and applications. Trans. Amer. Math. Soc. 267 (1981), 1–32.CrossRefGoogle Scholar
3Bahri, A. and Lions, P. L.. Remarques sur la théorie variationnelle des points critiques et applications. C. R. Acad. Sci. Paris 301 (1985), 145–7.Google Scholar
4Bahri, A. and Lions, P. L.. Morse index of some Min-Max critical points (Part I). Comm. Pure Appl. Math. 41 (1988), 1027–37.CrossRefGoogle Scholar
5Bahri, A. and Lions, P. L.. On the existence of positive solutions of semilinear elliptic equations in unbounded domains (Preprint).Google Scholar
6Benci, V. and Cerami, G.. Positive solution of some nonlinear elliptic problems in exterior domains. Arch. Rational Mech. Anal. 99 (1987), 283–300.CrossRefGoogle Scholar
7Bouligand, G.. Sur les fonctions de Green et de Neumann du cylindre. Bull. Soc. Math. France 42 (1914), 168–242.CrossRefGoogle Scholar
8Brezis, H. and Kato, T.. Remarks on the Schrodinger operator with singular complex potentials. J. Math. Pures Appl. 58 (1979), 137–51.Google Scholar
9Brezis, H. and Nirenberg, L.. Remarks on finding critical points. Comm. Pure Appl. Math. 44 (1991), 939–63.CrossRefGoogle Scholar
10Coron, J. M.. Topologie et cas limite des injections de Sobolev. C. R. Acad. Sci. Paris Ser. I 299 (1984), 209–12.Google Scholar
11Ekeland, I.. Nonconvex minimization problems. Bull. Amer. Math. Soc. 1 (1979), 443–74.CrossRefGoogle Scholar
12Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order (New York: Springer, 1983).Google Scholar
13Lien, W. C., Tzeng, S. Y. and Wang, H. C.. Existence of solutions of semilinear elliptic problems on unbounded domains. Differential Integral Equations 6 (1993), 1281–98.CrossRefGoogle Scholar
14Rabinowitz, P. H.. Multiple critical points of perturbed symmetric functional. Trans. Amer. Math. Soc. 272 (1982), 753–69.CrossRefGoogle Scholar
15Tanaka, K.. Morse indices and critical points related to the symmetric mountain pass theorem and applications. Comm. Parial Differential Equations 14(1) (1989), 99–128.CrossRefGoogle Scholar
16Zhu, X. P.. A perturbation result on positive entire solutions of a semilinear elliptic equation. J. Differential Equations 92 (1991), 163–78.CrossRefGoogle Scholar