Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T07:08:24.616Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains

Published online by Cambridge University Press:  14 November 2011

Xi-Ping Zhu  and
Huan-Song Zhou
Xi-Ping Zhu
Affiliation:
Department of Mathematics, Zhongshan University, Guangzhou 510275, PR China
Huan-Song Zhou
Affiliation:
Wuhan Institute of Mathematical Sciences, Academia Sinica, PO Box 71007, Wuhan 430071, PR China
Get access Rights & Permissions [Opens in a new window]

Synopsis

By using the concentration-compactness method of Lions [14, 16] and the mountain pass theorem of Ambrosetti and Rabinowitz [3], through a careful inspection of the energy balance for some sequence of approximated solutions, we show that under suitable conditions on f and h, the inhomogeneous problem. −Δu + c2u = λ(f(u) + h(x)) for x ∈ Ω (Ω is an exterior domain in ℝN, N≧ 3) and has at least two positive solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 115 , Issue 3-4 , 1990 , pp. 301 - 318
Copyright
Copyright © Royal Society of Edinburgh 1990

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 Space (New York: Academic Press, 1975).Google Scholar
2Amann, H..Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review 18 (1976), 620–709.CrossRefGoogle Scholar
3Ambrosetti, A. and Rabinowitz, P. H..Dual variational method in critical point theory and applications. J Fund. Anal. 14 (1973), 349–381.CrossRefGoogle Scholar
4Bahri, A. and Berestycki, H..A perturbation method in critical point theory and applications. Trans. Amer. Math. Soc. 267 (1981), 1–32.CrossRefGoogle Scholar
5Bahri, A. and Lions, P. L..Remaques sur la théorie variationnelle des points critiques et applications. C. R. Acad. Sci. Paris. 301 (1985), 145–147.Google Scholar
6Bahri, A. and Lions, P. L.. On the existence of a positive solution of semilinear elliptic equations in unbounded domains (preprint).Google Scholar
7Benci, V. and Cerami, G..Positive solutions of some nonlinear elliptic problems in exterior domains. Arch. Rational Mech. Anal. 99 (1987), 283–300.CrossRefGoogle Scholar
8Berestycki, H. and Lions, P. L..Nonlinear scalar field equations. I. Existence of a ground state; II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82 (1983), 313–375.CrossRefGoogle Scholar
9Brezis, H. and Nirenberg, L..Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), 437–477.CrossRefGoogle Scholar
10Crandall, M. G. and Rabinowitz, .P. H..Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Rational Mech. Anal. 58 (1975), 207–218.CrossRefGoogle Scholar
11Dautray, R. and Lions, J. L.. Analyse mathématique et calcul numerique pour les sciences et les techniques, tome 1, (Paris: Masson, 1984).Google Scholar
12Ding, W.-Y and Ni, W.-M.On the existence of positive entire solution of a semilinear elliptic equation. Arch. Rational Mech. Anal. 91 (1986), 283–308.CrossRefGoogle Scholar
13Ekland, I..Non-convex minimization problems. BAMS 1 (1979), 443–474.CrossRefGoogle Scholar
14Lions, P. L..On positive solutions of semilinear elliptic equations in unbounded domains. In Nonlinear Diffusion Equations and Their Equilibrium States, eds. Ni, W.-M, Peletier, L. A. and Serrin, J. (New York/Berlin: Springer, 1988).Google Scholar
15Lions, P. L..On the existence of positive solutions of semilinear elliptic equations. SIAM Review 24 (1982), 441–467.CrossRefGoogle Scholar
16Lions, P. L..The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 and part 2. Ann. Inst. H. Poincaré Anal. Nonlineaire 1 (1984), 109–145; 223–283.CrossRefGoogle Scholar
17Rabinowitz, P. H..Multiple critical points of perturbed symmetric functionals. Trans. Amer. Math. Soc. 272 (1982), 753–769.CrossRefGoogle Scholar
18Strauss, W..Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), 149–162.CrossRefGoogle Scholar
19Stuart, C. A..Bifurcation in L P(ℝN) for a semilinear elliptic equation. Proc. London Math. Soc. 57 (1988), 511–541.CrossRefGoogle Scholar
20Zhu, X.-P.. A perturbation result on positive entire solutions of a semilinear elliptic equation (preprint)Google Scholar
21Zhu, X.-P. and Cao, D.-M.. The concentration-compactness principle in nonlinear elliptic equation. Acta Math. Sci. 9 (1989).CrossRefGoogle Scholar